\section{Introduction}

Realistic quantum systems inherently interact with their surroundings and can be generally modeled by the open quantum dynamics. In the weak coupling limit between the system and the environment, the dynamics would be Markovian and described by the so-called quantum Markov semigroup (QMS) or the Lindblad equation, which is a natural analog of the Fokker-Planck equation in the quantum setting \cite{breuer2002theory}. Similarly to the theory of Markov semigroups, the analysis of the mixing time
is of central importance for a  QMS, and is closely related to the functional inequalities. 
In this work, we are interested in a class of convex Sobolev inequalities, referred to as quantum Beckner's inequalities. We will investigate their main properties and relations with other known quantum functional inequalities, such as Poincar\'{e} and modified log-Sobolev inequalities, via both algebraic and geometric approaches.  

\subsection{Classical convex Sobolev inequality} To motivate this work, we first review the results on the convex Sobolev inequalities in the classical setting. 
Let $(P_t)_{t \ge 0}$ be the symmetric diffusion semigroup associated with a Markov process $(X_t)_{t \ge 0}$ on a Riemannian manifold $M$ with metric $g(\cdot,\cdot)$ that has a unique invariant measure $\pi$. We denote by $L$ the generator of $P_t$ and define the Dirichlet form $\mc{E}(f,g)  := - \pi [f L g]$ for functions $f$ and $g$. Bakry and \'{E}mery in their seminal work \cite{bakry1985diffusions} showed that if there exists $\kappa > 0$ such that for $f \ge 0$,
\begin{align} \label{eq:curvature_dimension}
    \Gamma_2(f) \ge \kappa \Gamma (f)\,,
\end{align}
then the convex Sobolev inequality holds:
\begin{align} \label{eq:convex_ineq}
 2 \kappa \Ent_\pi^\phi(f) \le \mc{E}(\phi'(f),f)\,,
\end{align}
which is equivalent to the exponential decay of the $\phi$-entropy $\Ent_\pi^\phi(f) := \pi[\phi(f)] - \phi(\pi[f])$ and characterizes the convergence rate of the Markov process towards its invariant measure. Here $\Gamma$ and $\Gamma_2$ are carr\'{e} du champ operators \cite{bakry2014analysis}; $\phi:[0,\infty) \to \mathbb{R}$ is assumed to be
a smooth convex function such that $\phi(1) = \phi'(1) = 0$ and $1/\phi''$ is concave. In the cases: $\phi_1(s) = s(\log s - 1) + 1$ and $\phi_2(s) = s^2 - 2 s + 1$, up to some constant,  \eqref{eq:convex_ineq} gives the well-known modified log-Sobolev inequality (MLSI) and the Poincar\'{e} inequality, respectively, 
\begin{align} \label{ineq_a}
\alpha \big(\pi[f \log f] - \pi[f] \log \pi[f]\big) \le \mc{E}(\log f, f)\,,
\end{align}
and 
\begin{align} \label{ineq_b}
    \lambda \big(\pi[f^2] - \pi[f]^2 \big) \le \mc{E}(f , f)\,.
\end{align}
If we consider the interpolating family $\phi_p(s) = (s^p - s)/(p - 1) - s + 1$, $1 < p \le 2$, between $\phi_1$ and $\phi_2$, we obtain the Beckner's inequality:
\begin{equation} \label{ineq_c}
    \alpha_p \big( \pi[f^p] - \pi[f]^p \big)  \le p \mc{E}(f^{p - 1}, f)\,.
\end{equation}
Moreover, note from the diffusion property: $L \psi(f) = \psi'(f)L f + \psi''(f) \Gamma f$ for suitably smooth functions $\psi$ and $f$, that $\mc{E}(\log f , f) = 4 \mc{E}(f^{1/2}, f^{1/2})$ and $\mc{E}(f^q, f^{2 - q}) = (2 q - q^2) \mc{E}(f,f)$. By substituting $f = g^2$ and $f = g^q$ with $q = 2/p$ into \eqref{ineq_a} and \eqref{ineq_c}, respectively, up to constants, we have the usual log-Sobolev inequality (LSI):
\begin{equation} \label{ineq_d}
    \beta \big(\pi[g^2 \log g^2] - \pi[g^2] \log \pi[g^2]\big) \le  \mc{E}(g, g)\,,
\end{equation}
and the original Beckner's inequality first introduced in \cite{beckner1989generalized} for the Gaussian measure on $\mathbb{R}^n$:
\begin{equation} \label{ineq_e}
     \beta_q \big( \pi[g^2] - \pi[g^q]^{2/q} \big)  \le (2-q) \mc{E}(g, g)\,.
\end{equation}
The condition \eqref{eq:curvature_dimension} admits a deep geometric interpretation, and it is called the
Bakry-\'{E}mery (curvature-dimension) condition or $\Gamma_2$-criterion. To make this point clearer, let $L = \Delta_g - \nabla W \cdot \nabla$ be the generator associated with the Ornstein–Uhlenbeck process on the manifold $M$ that admits an invariant measure $d \pi = e^{- W} d \text{vol}_M$, where $W$ is the potential and $\text{vol}_M$ is the volume form on $M$. With the help of Bochner's formula, we can compute $\Gamma(f) = |\nabla f|^2$ and
$\Gamma_2(f) = |\nabla^2 f|^2 + \Ric(L)(\nabla f , \nabla f)$, where $\Ric(L)$ is the Ricci tensor for the generator $L$, defined by $\Ric(L): = \Ric_g + \nabla^2 W$ with $\Ric_g$ being the standard Ricci curvature of $M$. It is easy to prove that the condition \eqref{eq:curvature_dimension} holds if and only if the Ricci curvature of $L$ is bounded below: $\Ric(L)(\cdot,\cdot) \ge \kappa g(\cdot,\cdot)$. Otto-Villani \cite{otto2000generalization} and von Renesse-Sturm \cite{von2005transport} further observed that \eqref{eq:curvature_dimension} is also equivalent to that the relative entropy with respect to $d \pi$ is displacement $\kappa$-convex on the Wasserstein space of probability measures on $M$. Inspired by this characterization, Sturm \cite{sturm2006geometry} and Lott-Villani \cite{lott2009ricci} extended the notion of Ricci curvature 
to metric measure spaces by exploiting the convexity properties of entropy functionals. See \cite{bakry2014analysis,villani2009optimal} for more details.

The above framework establishes a beautiful connection between various subjects such as partial differential equations (PDE), probability, and geometry, and has led to important research progress in these fields. The key step in the Bakry-\'{E}mery arguments \cite{bakry1985diffusions} is to estimate the second derivative of the relative entropy along the Markov semigroup, where the calculation depends on  Bochner’s formula or, more abstractly, the diffusion property. Arnold et al. \cite{arnold2001convex,arnold2008large,arnold2014sharp} revisited the Bakry-\'{E}mery method in the PDE framework and characterized the long-time asymptotics for various classes of Fokker-Planck type equations based on the convex Sobolev inequalities; see also \cite{otto2001geometry,del2002best,carrillo2000asymptotic,markowich2000trend} for the applications of functional inequalities in nonlinear Fokker-Planck type equations. Among the general convex Sobolev inequalities, the Beckner's inequality is of particular interest, since it provides an interpolating family between MLSI and Poincar\'{e} inequality. The recent work \cite{gentil2021family} proved a class of weighted Beckner's inequalities and the refined ones based on the Bakry-\'{E}mery method and the curvature-dimension conditions.  We also mention that \cite{arnold2007interpolation} proved the inequality \eqref{ineq_e} by  the hypercontractivity and spectral estimates. In particular, Dolbeault et al. \cite{dolbeault2009new,dolbeault2012poincare} explored the gradient flow structure of the Fokker-Planck equation for general entropy functionals and proved the contraction of the associated transport distance along the Fokker-Planck flow, which gave a unified  gradient flow framework for investigating the convex Sobolev inequalities \eqref{eq:convex_ineq}. 

It is also desirable to extend the theory of convex Sobolev inequalities to the setting of finite Markov chains. In this case, due to the lack of chain rule, the inequalities \eqref{ineq_a} and \eqref{ineq_d}, also, \eqref{ineq_c} and \eqref{ineq_e} are not equivalent (one is stronger than the other) \cite{bobkov2006modified}. For instance, Dai Pra et al.  \cite{dai2002entropy} provided an example where the MLSI \eqref{ineq_a} holds while the LSI \eqref{ineq_d} fails.  In what follows, to avoid confusion between \eqref{ineq_c} and \eqref{ineq_e}, following \cite{adamczak2022modified} and \cite{chafai2004entropies} we call the inequality of the form \eqref{ineq_e} the dual Beckner's inequality. 
Similarly to the diffusion case, the Bakry-\'{E}mery method and gradient flow techniques are two main approaches for the validity of \eqref{eq:convex_ineq}. J\"{u}ngel and Yue \cite{jungel2017discrete} followed Bakry-\'{E}mery's ideas and gave the conditions of $\phi$ under which \eqref{eq:convex_ineq} holds. The proof 
    relies on a discrete Bochner-type identity that was first introduced in \cite{boudou2006spectral,caputo2009convex}. Recently, Weber and Zacher \cite{weber2021entropy} proposed discrete analogs of the condition \eqref{eq:curvature_dimension} such that the MLSI \eqref{ineq_a} and Beckner’s inequality \eqref{ineq_c} hold. Their argument, different from \cite{jungel2017discrete}, is based on the modified $\Gamma$ and $\Gamma_2$ operators that satisfy some kind of discrete diffusion property. 
    We point out that a probabilistic approach, based on the Bakry-\'{E}mery method and the coupling arguments, for the discrete convex Sobolev inequalities can be found in \cite{conforti2022probabilistic} by Conforti. 
    
    The starting point of the gradient flow approach for discrete functional inequalities is \cite{maas2011gradient} where Maas defined a discrete transport distance such that the continuous time finite Markov chains can be identified as the gradient flow of the relative entropy. Following the ideas of \cite{lott2009ricci,sturm2006geometry}, Erbar and Maas \cite{erbar2012ricci} introduced the discrete Ricci curvature based on this discrete Wasserstein metric, and derived a number of functional inequalities including the discrete MLSI and the transport cost inequalities; see also \cite{erbar2018poincare}. Later, Fathi and Maas \cite{fathi2016entropic} generalized the discrete Bochner formula \cite{caputo2009convex} and developed a systematic approach for estimating the discrete Ricci curvature lower bounds. It is worth mentioning that both the discrete Bakry-\'{E}mery condition in \cite{weber2021entropy} and the discrete Ricci curvature in \cite{erbar2012ricci} enjoy the tensorization properties, and the aforementioned general results can be applied to several interesting models such as birth-death processes, random transposition models and Bernoulli-Laplace models (see related papers for details). We refer the readers to the review \cite{maas2017entropic} and the references therein for other notions of the Ricci curvature in the discrete setting and their implications on functional inequalities. 

\subsection{Quantum functional inequalities} 
In analogy with the classical case, quantum functional inequalities play a fundamental role in understanding the asymptotic behavior of a QMS. The study of LSI in the noncommutative setting may date back to \cite{gross1975hypercontractivity},
and its connections with the hypercontractivity were fully discussed in the seminal work \cite{olkiewicz1999hypercontractivity}. 
The quantum MLSI was first introduced by Kastoryano and Temme \cite{kastoryano2013quantum} for deriving improved bounds on the mixing time of 
primitive quantum Markov processes than those obtained in \cite{temme2010chi} via the Poincar\'{e} inequality. The quantum MLSI constant was investigated in detail in \cite{muller2016entropy,muller2016relative} for some special models, e.g., the depolarizing semigroup and the doubly stochastic qubit Lindbladian. Regarding the general framework for the validity of quantum MLSI, the Bakry-\'{E}mery method, 
which has been successfully adapted to the discrete case, (seems) has not been explored in the literature for the quantum semigroups and needs further research. However, the notion of Ricci curvature lower bounds (geodesic convexity) in the quantum setting, pioneered by Carlen and Maas \cite{carlen2014analog,carlen2017gradient}, has shown its utility in proving the quantum MLSI and related functional inequalities with numerous applications in concrete physical models. We briefly outline the main progress in this direction below. Carlen and Maas \cite{carlen2017gradient} introduced a quantum analog of 2-Wasserstein distance such that the primitive QMS satisfying the $\sigma$-detailed balance condition (cf.\,Definition \ref{def:sidbc}) can be written as the gradient flow of the relative entropy and showed that the relative entropy is geodesically convex for  the Fermi and Bose Ornstein-Uhlenbeck semigroups, which extended their previous work \cite{carlen2014analog}. Based on Carlen and Maas's results, Datta and Rouz{\'e} \cite{rouze2019concentration,datta2020relating} considered the Ricci curvature of a QMS, and obtained some quantum Sobolev and concentration inequalities, generalizing the results in the classical regime \cite{otto2001geometry,erbar2012ricci,erbar2018poincare}. Wirth and Zhang \cite{wirth2021curvature} further introduced noncommutative curvature-dimension conditions and derived some dimension-dependent quantum functional inequalities.   

One of the favorable features of the classical LSI is the tensorization property, which enables obtaining the functional inequalities for the tensor product systems from those for the subsystems. However, this property is known to fail for the quantum MLSI (cf.\cite[Proposition 4.21]{brannan2022complete}). To circumvent such difficulty, Gao et al. \cite{gao2020fisher} introduced the complete modified log-Sobolev inequality (CMLSI), which is a stronger notion than the quantum MLSI,
and showed that it satisfies the desired tensorization properties. Very recently, Gao and Rouz\'{e} proved, by a two-sided estimate for the relative entropy, that the CMLSI holds for any finite dimensional $\sigma$-detailed balance QMS. We refer the interested readers to \cite{brannan2022complete,li2020graph,brannan2021complete} for 
the geometrical approaches for studying the CMLSI, which is based on the noncommutative Ricci curvature and the group transference techniques.

\subsection{Main results and layout} As reviewed above, although there has been much progress on Beckner's inequalities in both diffusion and discrete cases, the results for quantum Beckner's inequalities are quite limited. We only note the recent work by Li \cite{li2020complete} where the author investigated the matrix-valued Beckner's inequalities, in terms of the Bregman relative entropy \cite{molnar2016maps}, for symmetric semigroups on a finite von Neumann algebra. In this work we try to close this gap. We consider the primitive QMS with $\sigma$-detailed balance condition for $\sigma \in \mathcal{D}_+(\mc{H})$, and define the family of quantum $p$-Beckner's inequalities and their dual version, by extending the definitions in \cite{adamczak2022modified,latala2000between,chafai2004entropies} for classical Markov semigroups; see Definition \ref{def:quantum_func} for the functional inequalities that we will mainly focus on.
It turns out that the $p$-Beckner's inequality \eqref{ineq_becp} describes the rate at which the quantum $p$-divergence $\mc{F}_{p,\sigma}$ \eqref{def:quanpdivi}, which can be viewed as the normalized noncommutative $L_p$-norm, tends to zero. The diagram in Figure \ref{fig:diagram} below summarizes the main results of Sections \ref{sec:inter_primi} and \ref{sec:positive_stab}. 

\begin{figure}[h]
\begin{tikzcd}[row sep= 2.5 em , column sep= 5 em]
\eqref{ineq_becp} \arrow[Leftrightarrow]{dd}[swap]{\text{Theorem}\ \ref{thm:beck_log_sobo}} \arrow[Leftrightarrow]{rd}{\text{Theorem}\ \ref{thm:beck_poincare},\ \text{Corollary}\ \ref{coro:pitobeck} } & &
\eqref{ineq_dbecq} \arrow[Leftrightarrow]{dd}{\text{Proposition} \ \ref{dualbecktolsi}} \arrow[Rightarrow]{ll}[swap]{\text{Proposition} \ \ref{prop:beck_to_dual_beck}} \\
&  \eqref{ineq_pi} \arrow[Leftrightarrow]{rd}[swap]{\text{\cite[Corollary 6]{temme2014hypercontractivity}}}       &    \\
\eqref{ineq_mlsi} \arrow[Leftrightarrow]{ur}{\text{\cite{gao2021complete,cao2019gradient}}}  && \eqref{ineq_lsi}  \arrow[Rightarrow]{ll}{\text{\cite[Proposition 13]{kastoryano2013quantum}}}
\end{tikzcd}
\caption{Chain of quantum convex Sobolev inequalities.}
\label{fig:diagram}
\end{figure}

We first note that the relations between \eqref{ineq_pi}, \eqref{ineq_mlsi}, and \eqref{ineq_lsi} have been well investigated. \cite[Corollary 6]{temme2014hypercontractivity} provided a two-sided bound for the log-Sobolev constant by the Poincar\'{e} constant; \cite{kastoryano2013quantum} showed that the modified log-Sobolev constant is bounded below by the log-Sobolev constant and above by the Poincar\'{e} constant. This also allows us to conclude that \eqref{ineq_mlsi} is equivalent to \eqref{ineq_pi} for the primitive QMS; see also \cite[Proposition I.7]{cao2019gradient} and \cite[Theorem 3.3]{gao2021complete}. Here, in Section \ref{sec:inter_primi}, we prove that the dual  $q$-Beckner's inequalities \eqref{ineq_dbecq}, as an interpolating family between \eqref{ineq_lsi} and \eqref{ineq_pi}, is equivalent to \eqref{ineq_lsi} in the sense of Proposition \ref{dualbecktolsi}. We also show in Proposition \ref{prop:beck_to_dual_beck} that \eqref{ineq_dbecq} is stronger than \eqref{ineq_becp}. Propositions \ref{prop:mono_dual_beck} and \ref{prop:mono_beck} discuss the monotonicity of the dual Beckner constant $\beta_q(\mc{L})$ and the Beckner constant $\alpha_p(\mc{L})$, respectively. Then, in Section \ref{sec:positive_stab}, we investigate the quantum Beckner's inequalities in detail. Inspired by the work \cite{gao2021complete}, we show in Lemma \ref{lem:two-sided} that the quantum $p$-divergence $\mc{F}_{p,\sigma}$ can be bounded from above and below by the $\chi_{\kappa_{1/p}}^2$-divergence with $\kappa_{1/p}$ being the power difference. This key lemma enables us to derive a two-sided estimate for the Beckner constant $\alpha_p(\mc{L})$ in terms of the Poincar\'{e} constant $\lambda(\mc{L})$ in a similar way as \cite[Theorem 3.3]{gao2021complete}; see Theorem \ref{thm:beck_poincare}. We proceed to prove, by contradiction, in Theorem \ref{thm:beck_log_sobo} that for each $p \in (1,2]$, \eqref{ineq_becp} is equivalent to \eqref{ineq_mlsi}, and there holds $\alpha_p(\mc{L}) \to \alpha_1(\mc{L})$ as $p \to 1^+$, where $\alpha_1(\mc{L})$ is the modified log-Sobolev constant. This extends the result \cite[Theorem 1.1]{adamczak2022modified} in the classical setting. We also extend the main result in \cite{junge2019stability} and provide a stability estimate for the Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state $\sigma$; see Theorem \ref{thm:beck_stab}. In Section \ref{sec:app_exp}, we first show in Proposition \ref{prop:mixing} that the family of $p$-Beckner's inequalities may imply a better mixing time bound than the ones in \cite{kastoryano2013quantum,temme2010chi}. In Proposition \ref{prop:moment}, we extend \cite[Proposition 3.3]{adamczak2022modified} for the classical case to the quantum setting and obtain moment estimates under \eqref{ineq_becp}. Then we analyze the quantum Beckner constant for the depolarizing semigroup and reduce it to the computation of the classical one for a Markov chain on the two-point space; see Proposition \ref{prop:beck_depol}. 


Another motivation for the current work is \cite{dolbeault2009new,dolbeault2012poincare}, where the authors provided a gradient flow approach for the classical Beckner's inequalities. To be precise, 
Dolbeault et al. \cite{dolbeault2009new} defined the following class of transport distances $\mc{W}_{2,\alpha,\gamma}$ with $\alpha \in [0,1]$ \`{a} la Benamou-Brenier: for probability measures $\mu_0$ and $\mu_1$ on $\mathbb{R}^d$, 
\begin{align} \label{def:distance_savare}
\mc{W}_{2,\alpha,\gamma}(\mu_0,\mu_1): = \inf\Big\{
\int_0^1\int_{\mathbb{R}^d} \rho_t^{-\alpha}|w_t|^2\ d \gamma dt\,;\ \partial_t \mu_t + \nabla \cdot {\bf \nu}_t = 0\,,\ \mu_t = \rho_t \gamma + \mu_t^\perp\,,\ \nu_t = w_t \gamma \ll \gamma 
\Big\}\,,
\end{align}
where $\gamma$ is a reference Radon measure. In the  case $\alpha = 1$, $\mc{W}_{2,\alpha,\gamma}(\mu_0,\mu_1)$ gives the quadratic Wasserstein distance \cite{benamou2000computational}, while when $\alpha = 0$, it is equivalent to the weighted homogeneous $\dot{H}_\gamma^{-1}$ Sobolev distance \cite{peyre2018comparison,villani2003topics}:
\begin{align*}
    \norm{\mu_0 - \mu_1}_{\dot{H}_\gamma^{-1}} = \sup \Big\{\int_{\mathbb{R}^d} \xi\ d(\mu_0 - \mu_1)\,;\ \xi \in C_c^1(\mathbb{R}^d)\,,\ \int_{\mathbb{R}^d} |\nabla \xi|^2\ d \gamma \le 1  \Big\}\,.
\end{align*}
Thus $\mc{W}_{2,\alpha,\gamma}$ can be viewed as a natural interpolating family between them. Moreover, let the reference measure be $\gamma := e^{-V} \mathscr{L}^d $ with the potential $V$ being smooth and convex, where $\mathscr{L}^d$ is the Lebesgue measure on $\mathbb{R}^n$.
With such choice of $\gamma$, they showed that the gradient flow of the (Tsallis) functional, for $\alpha \in [0,1)$,
\begin{align} \label{def:class_func}
    \mathscr{F}_\alpha(\mu): = \frac{1}{(2-\alpha)(1-\alpha)} \int_{\mathbb{R}^d} \rho^{2-\alpha}\  d\gamma\,,\quad \mu = \rho \gamma\,,
\end{align}
is the Fokker-Planck equation: 
\begin{align*}
\partial_t \mu - \Delta \mu - \nabla \cdot (\mu \nabla V) = 0\,.
\end{align*}
In the subsequent work \cite{dolbeault2012poincare}, they further proved that $\mc{F}_\alpha$ is geodesically $\lambda$-convex under the assumption: $\nabla^2 V \ge \lambda I$ for $\lambda > 0$, which implies the classical Beckner's inequality. 


It is easy to note that in the commutative setting, up to some constant, our quantum $p$-divergence $\mc{F}_{p,\sigma}$ \eqref{def:quanpdivi} is nothing else but the functional $ \mathscr{F}_\alpha$ in \eqref{def:class_func} with $p = 2 - \alpha$.  In Section \ref{sec:qot_beck}, we extend the results in \cite{dolbeault2009new,dolbeault2012poincare}
to the quantum regime and provide a geometric characterization for the quantum $p$-Beckner's inequality. To do so, we first construct a Riemannian metric $g_{p,\rho}$ on the quantum states in Section \ref{sec:gradient}, so that the QMS with $\sigma$-detailed balance is  the gradient flow of the quantum $p$-divergence $\mc{F}_{p,\sigma}$ with respect to $g_{p,\rho}$. Then in Section \ref{sec:quantum_distance}, we investigate the properties of the associated Riemannian distance, denoted by $W_{2,p}$ (cf.\,\eqref{def:wp1}), which can be regarded as a quantum analog of $W_{2,\alpha,\gamma}$ in \eqref{def:distance_savare}. The main result in this section is Theorem \ref{thm:main_wasser}, where we show that $(\mathcal{D}(\mc{H}),W_{2,p})$ is a complete geodesic metric space. We also prove in Proposition \ref{prop:repkernel} that similarly to the classical case,
the new class of distances $W_{2,p}$ is an interpolating family between the quantum 2-Wasserstein distance defined by Carlen and Maas \cite{carlen2017gradient} and the noncommutative $\dot{H}^{-1}$ Sobolev distance \eqref{eq:distance_1}. With these results, it is straightforward to define the entropic Ricci curvature associated with the functional $\mc{F}_{p,\sigma}$ 
in the spirit of \cite{sturm2006geometry,lott2009ricci,datta2020relating}. We then derive an HWI-type interpolation inequality from the Ricci curvature lower bound and show that the positive Ricci curvature can imply the quantum Beckner's inequalities. Furthermore , we prove the 
following chain of quantum functional inequalities:
\begin{equation} \label{eq:chain}
\eqref{ineq_becp}   \xLongrightarrow{\text{Proposition}\  \ref{propa}}   \eqref{ineq:tc} \xLongrightarrow{\text{Proposition}\  \ref{propb}}  \eqref{ineq_pi}\,,
\end{equation} 
where \eqref{ineq:tc} is a transport cost inequality associated with $W_{2,p}$. These results are presented in Section \ref{sec:geodesic_convexity}.



We end the introduction  with a list of notations used throughout this work:
\begin{enumerate}[\textbullet]
    \item Let $\mc{B}(\mc{H})$ denote the space of bounded operators on a complex Hilbert space $\mc{H}$ of dimension $d < \infty$. $\mc{B}_{sa}(\mc{H})$ is the subspace of self-adjoint operators on $\mc{H}$, while $\mc{B}^{+}_{sa}(\mc{H})$ is the cone of positive semidefinite operators.  For simplicity, in what follows, by $A \ge 0$ (resp., $A > 0$) we mean a positive semidefinite (resp., definite) operator.
    \item We denote by $\langle\cdot,\cdot\rangle$ the Hilbert-Schmidt inner product on $\mathcal{B}(\mc{H})$, i.e., $\langle X, Y\rangle = \tr (X^*Y)$, where $X^*$ is the adjoint operator of $X$. Moreover, we write $\Phi^\dag$ for the adjoint of a superoperator $\Phi: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ with respect to the inner product $\langle \cdot,\cdot \rangle $. 
    The modulus of $X \in \mathcal{B}(\mc{H})$ is defined by $|X|: = \sqrt{X^* X} $. 
    \item We define the Schatten $p$-norm by $\norm{X}_p = (\tr(|X|^p))^{1/p}$ for $X \in \mathcal{B}(\mc{H})$ and $p \in [1,\infty]$. In particular, $\norm{\cdot}_\infty$ is the operator norm. For a superoperator on $\mathcal{B}(\mc{H})$, the operator norm is simply denoted by $\norm{\cdot}$.
    \item We denote by $\mc{D}(\mc{H}) := \{\rho \in \mc{B}_{sa}^+(\mc{H})\,;\  \tr \rho =1 \}$ the set of density operators (quantum states), and by $\mc{D}_+(\mc{H})$ the full-rank density operators.
   
    \item The identity operator on $\mc{H}$ is written as ${\bf 1}_{\mc{H}}$ (or ${\bf 1}$, when there is no confusion). Similarly, the identity superoperator on $\mathcal{B}(\mc{H})$ is denoted by ${\rm id}_{\mc{H}}$, or simply ${\rm id}$.
    \item Let $\mc{M}$ be a subset of $\mc{B}(\mc{H})$. We denote by $\mc{M}^J$ the set of vector fields over $\mc{M}$, i.e., ${\bf A} = (A_1, \cdots, A_J)  \in \mc{M}^J$ for $A_j \in \mc{M}$, $1 \le j \le J$. The Hilbert-Schmidt inner product     naturally extends to $\mc{M}^J$ as $\langle {\bf A} , {\bf B} \rangle = \sum_{j = 1}^J \langle A_j, B_j\rangle$. 
    \item We shall always write $\h{p}$ for the H\"{o}lder conjugate for $ 1 \le p \le \infty$, i.e., $1/p + 1/\h{p} = 1$. 
\end{enumerate}


 

\section{Preliminaries}



\subsection{Quantum Markov semigroup}


Let us first recall preliminaries about the Markovian open quantum dynamics. We say that $(\mc{P}_t)_{t \ge 0}: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ is a quantum Markov semigroup if $\mc{P}_t$ is a $C_0$-semigroup of completely positive, unital maps, whose generator $\mc{L}$ is called the Lindbladian, defined by $\mc{L}(X): = \lim_{t \to 0} t^{-1}(\mc{P}_t (X) - X)$. 
A quantum channel $\Phi: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})$ is a completely positive trace preserving (CPTP) map. Then, we see that the dual QMS $\mc{P}_t^\dag$ is a semigroup of quantum channels. The associated equation $\partial_t \rho = \mc{L}^\dag \rho$ is referred to as the Lindblad equation. In this work, we shall always consider the QMS $\mc{P}^\dag_t$ that 
admits a unique full-rank invariant state $\sigma$, that is, there exists a unique $\sigma \in \mc{D}_+(\mc{H})$ such that $\mc{P}_t^\dag(\sigma) = \sigma$ for $t \ge 0$. In this case, the QMS $\mc{P}_t$ is said to be primitive; and it holds that \cite{frigerio1982long}
\begin{align} \label{eq:conver_qms}
    \lim_{t \to \infty} \mc{P}_t(X) = \tr(\sigma X){\bf 1}\quad \text{for} \ X \in \mathcal{B}(\mc{H})\,. 
\end{align}


For $A \in \mathcal{B}(\mc{H})$ we define the left and right multiplication operator on $\mathcal{B}(\mc{H})$ by $L_A = A X$ and $R_A = X A$, respectively. It is easy to see that $L_{f(A)} = f(L_A)$ and $R_{f(A)} = f(R_A)$ holds for $A > 0$ and 
functions $f: (0,\infty) \to \mathbb{R}$. We also introduce the relative modular operator $\Delta_{\rho,\sigma} = L_\rho R_\sigma^{-1}: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ for 
$\rho,\sigma \in \mc{D}_+(\mc{H})$. When $\sigma = \rho$, we simply write it as $\Delta_\sigma$. We next introduce the quantum detailed balance condition (DBC). For this, we define a family of inner products on $\mathcal{B}(\mc{H})$: for a given $\sigma \in \mc{D}_+(\mc{H})$ and $s \in \mathbb{R}$, 
\begin{align} \label{def:s-inner}
    \langle X, Y\rangle_{\sigma,s} := \tr(\sigma^s X^* \sigma^{1-s} Y) = \langle X, \Delta_\sigma^{1-s} Y \sigma \rangle\,,
\end{align}
where $\langle \cdot, \cdot\rangle_{\sigma,1}$ and $\langle \cdot, \cdot \rangle_{\sigma,1/2}$ are called the GNS and KMS inner products, respectively. In particular, when $\sigma$ is the maximally mixed state ${\bf 1}/d$, all the inner products $\langle \cdot,\cdot \rangle_{\sigma,s}$ reduce to the normalized Hilbert-Schmidt inner product: 
\begin{align*}
    \langle X , Y\rangle_{\frac{{\bf 1}}{d}} := \frac{1}{d} \langle X , Y\rangle\,,\quad \forall X, Y \in \mathcal{B}(\mc{H})\,.
\end{align*}



\begin{definition} \label{def:sidbc}
We say that a QMS $\mc{P}_t$, or its generator $\mc{L}$, satisfies the $\sigma$-DBC for some $\sigma \in \mc{D}_+(\mc{H})$ if the operator $\mc{L}$ is self-adjoint with respect to the inner product $\langle\cdot, \cdot\rangle_{\sigma,1}$, that is,
\begin{equation*}
    \langle \mc{L}(X), Y\rangle_{\sigma,1} = \langle X, \mc{L}(Y)\rangle_{\sigma,1}\,,\quad \forall X, Y \in \mc{B}(\mc{H})\,.
\end{equation*}
\end{definition}
One can readily see that $\mc{P}_t$ is symmetric, i.e., $\mc{P}_t = \mc{P}_t^\dag$, if and only if it satisfies $\sigma$-DBC for the maximally mixed state $\sigma = {\bf 1}/d$, and that if $\mc{P}_t$ satisfies $\sigma$-DBC, then $\sigma$ is an invariant state of $\mc{P}_t$. More generally, for any function $f: (0,\infty) \to (0,\infty)$ and $\sigma \in \mc{D}_+(\mc{H})$, we define the operator:
\begin{align} \label{def:operator_kernel}
    J_\sigma^f := R_\sigma f(\Delta_\sigma): \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align}
and the associated inner product:
\begin{align} \label{eq:general_inner}
    \langle X, Y\rangle_{\sigma,f} := \langle X, J_\sigma^f (Y)\rangle\,.
\end{align}
It is clear that $\langle X, Y\rangle_{\sigma,f}$ with $f = x^{1-s}$ gives the inner product \eqref{def:s-inner}; and the adjoint of a linear operator $\mc{K}$ on $\mathcal{B}(\mc{H})$ with respect to $\langle\cdot,\cdot\rangle_{\sigma,f}$ is given by $ ( J_\sigma^f)^{-1} \mc{K}^\dag   J_\sigma^f $.
The following result from \cite[Theorem 2.9]{carlen2017gradient} relates the self-adjointness of $\mc{L}$ with respect to different inner products. 



\begin{lemma} \label{lem:self_adjoint}
If a QMS $\mc{P}_t$ satisfies the $\sigma$-DBC for some state $\sigma \in \mc{D}_+(\mc{H})$, then its generator $\mc{L}$  
commutes with the modular operator $\Delta_\sigma$, and it is self-adjoint with respect to $\langle\cdot, \cdot\rangle_{\sigma,f}$ for any $f:(0,\infty) \to (0,\infty)$, i.e., 
\begin{align*}
    \mc{L}\Delta_\sigma = \Delta_\sigma \mc{L}\,,\quad   J_\sigma^f  \mc{L} = \mc{L}^\dag    J_\sigma^f\,.
\end{align*}
In particular, $\mc{L}$ is self-adjoint with respect to the KMS inner product $\langle \cdot,\cdot \rangle_{\sigma,1/2}$ and there holds
\begin{align*}
    \Gamma_\si \mc{L} = \mc{L}^\dag \Gamma_\si\,.
\end{align*}
\end{lemma}

The next lemma, due to Alicki \cite{alicki1976detailed}, characterizes the structure of the generator $\mc{L}$ of a QMS $\mc{P}_t$ satisfying $\sigma$-DBC; see also \cite{kossakowski1977quantum,carlen2017gradient}.



\begin{lemma}\label{lem:struc_gene}
Suppose that $\mc{P}_t$ satisfies $\sigma$-DBC for some $\sigma \in \mc{D}_+(\mc{H})$. Then it holds that 
 \begin{align} \label{eq:structure}
        \mc{L}(X) = \sum_{j = 1}^{J} (e^{-\omega_j/2}V_j^*[X,V_j] + e^{\omega_j/2}[V_j,X]V^*_j)\,,
    \end{align}
with $\omega_j \in \mathbb{R}$ and $J \le n^2 -1$. Here $V_j \in \mathcal{B}(\mc{H})$, $1\le j \le J$, are trace zero and orthogonal eigenvectors of $\Delta_\sigma$:
\begin{align} \label{eq:eigmodular}
    \Delta_\sigma(V_j) =  e^{-\omega_j}V_j\,,\quad \langle V_j, V_k\rangle = c_j\delta_{j,k}\,,\quad \tr(V_j) = 0\,,
\end{align}
where $c_j > 0$ are  normalization constants. Moreover, for each $1 \le j \le J$, there exists $1 \le  j' \le J $ such that 
\begin{align} \label{eq:adjoint_index}
    V_j^* = V_{j'}\,,\quad \omega_j = - \omega_{j'}\,.
\end{align}
\end{lemma}

The real numbers $\omega_j \in \mathbb{R}$ are called Bohr frequencies of the Lindbladian $\mc{L}$, which are uniquely determined by the invariant state $\sigma$, while the operators $V_j$ are only unique up to unitary transformations. In what follows, we will fix a set of $V_j$ for the representation \eqref{eq:structure} with the properties in Lemma \ref{lem:struc_gene}. 

\begin{remark} \label{eq:tracial}
When the invariant state $\sigma$ is the maximally mixed state ${\bf 1}/d$, we have that
$\Delta_\sigma$ is the identity operator and hence $\omega_j = 0$ by \eqref{eq:eigmodular}, and we can take the operators $V_j$ to be self-adjoint. In this symmetric case, the QMS $\mc{P}_t$ may be regarded as a noncommutative heat semigroup, and it generator has the form:
\begin{align} \label{eq:gen_symm}
    \mc{L}(X) = - \sum_j [V_j, [V_j, X]]\,.
\end{align}
\end{remark}


\begin{example} \label{exp:dep_channel}
An important example of QMS is the generalized depolarizing semigroup: for $\sigma \in \mathcal{D}_+(\mc{H})$ and $\gamma > 0$, 
\begin{align} \label{def:depolt}
    \mc{P}_t(X) = e^{- \gamma t} X + (1 - e^{- \gamma t}) \tr(\sigma X) {\bf 1}\,, \quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
generated by 
\begin{align} \label{def:depol}
    \mc{L}_{depol}(X) = \gamma  (\tr(\sigma X) {\bf 1} - X)\,.
\end{align}
It is easy to see that $\mc{P}_t$ is primitive and satisfies $\sigma$-DBC. 
\end{example}



Lemma \ref{lem:struc_gene} actually gives a useful first-order differential structure associated with the QMS with $\sigma$-DBC. We introduce the weighting operator for a full-rank quantum state 
$\sigma \in \mc{D}_+(\mc{H})$:  
\begin{align*}
    \Gamma_\si X = \sigma^{\frac{1}{2}}X\sigma^{\frac{1}{2}}:\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align*} 
and the noncommutative analog of partial derivatives (associated with the generator $\mc{L}$): 
\begin{align*}
    \partial_j X = [V_j, X]:\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,. 
\end{align*}
Then the noncommutative gradient $\nabla: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})^J$ and divergence $ {\rm div} : \mathcal{B}(\mc{H})^J \to \mathcal{B}(\mc{H})$ can be defined as 
\begin{equation*}
    \nabla X = (\partial_1 X, \cdots, \partial_J X) \quad \text{for} \ X \in \mathcal{B}(\mc{H})\,,
\end{equation*}
and 
\begin{equation*}
    {\rm div} {\bf X} = - \sum_{j = 1}^J \partial_j^\dag X_j  \quad \text{for} \ {\bf X} \in \mathcal{B}(\mc{H})^J\,,
\end{equation*}
respectively. By definition and \eqref{eq:eigmodular},
it follows that the adjoint of $\partial_j$ with respect to $\langle \cdot, \cdot  \rangle_{\sigma,1/2}$ is given by \begin{align} \label{eq:adjpjkms}
    \partial_{j,\sigma}^\dag B = \Gamma_\si^{-1}\partial_j^\dag \Gamma_\si B = e^{-\omega_j/2} V_j^* B - e^{\omega_j/2} B V_j^*\,.  
\end{align}
With the help of $\partial^\dag_{j,\sigma}$ \eqref{eq:adjpjkms}, we can rewrite \eqref{eq:structure} as 
\begin{align} \label{eq:rep_gen}
    \mc{L} (X) = - \sum_{j = 1}^J \partial_{j,\sigma}^\dag \partial_j X\,,
\end{align}
and a noncommutative integration by parts formula holds: 
\begin{align} \label{eq:inte_by_parts}
    - \langle Y, \mc{L}(X)\rangle_{\sigma,1/2} = \sum_{j = 1}^J \langle \partial_j Y, \partial_j X    \rangle_{\sigma,1/2} \quad \text{for} \ X, Y \in \mathcal{B}(\mc{H})\,.
\end{align}
The following lemma will be useful below, which generalizes \cite[Proposition 4.12]{carlen2020non}. 
\begin{lemma}\label{lem:DBC}
Suppose that $\mc{P}_t$ satisfies $\sigma$-DBC, then  for a continuous function $f:(0,\infty) \to (0,\infty)$, there holds 
\begin{align*}
    - \langle Y, \mc{L} X\rangle_{\sigma,f} = \sum_{j = 1}^J \left\langle \partial_j Y,  R_{e^{-\omega_j/2} \sigma} f\left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2} \sigma}\right) \partial_j X \right\rangle\,,\quad X, Y \in \mathcal{B}(\mc{H})\,.
\end{align*}
\end{lemma}
\begin{proof}
By Stone-Weierstrass theorem, it suffices to consider $f = x^s$. For this, we have 
\begin{align*}
    - \langle Y, R_\sigma \Delta_\sigma^s \mc{L} X\rangle = \sum_{j = 1}^J \langle Y, \Delta_\sigma^{s-\frac{1}{2}}\partial_j^\dag \Gamma_\si \partial_j X  \rangle & = \sum_{j = 1}^J  e^{(s-\frac{1}{2})\omega_j} \left\langle \partial_j Y, R_\sigma \Delta_\sigma^s \partial_j X \right\rangle \\
    & = \sum_{j = 1}^J  \left\langle \partial_j Y, R_{e^{-\omega_j/2}\sigma} \left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2}\sigma}\right)^s \partial_j X \right\rangle\,,
\end{align*}
by noting $
    \Delta_\sigma^s \partial_j X = e^{-s\omega_j} \partial_j \Delta_\sigma^s X
$ from \eqref{eq:eigmodular}.
\end{proof}


\subsection{Quantum entropy and Dirichlet form}


In this section, we shall introduce the notions of quantum relative entropies and Dirichlet forms, and discuss their basic properties.  We start with the definition of the noncommutative weighted $L_p$ space. For $p > 0$ and $\sigma \in \mathcal{D}_+(\mc{H})$, we define the $\sigma$-weighted $p$-norm (quasi-norm if $p < 1$) on $\mathcal{B}(\mc{H})$ \cite{olkiewicz1999hypercontractivity,kastoryano2013quantum}:
\begin{align*}
    \norm{X}_{p,\sigma} := \tr\Big(|\Gamma_\sigma^{1/p}(X)|^p \Big)^{1/p}\,.
\end{align*}
In particular, if $\sigma = \frac{{\bf 1}}{d}$, then $\norm{\cdot}_{p,\frac{{\bf 1}}{d}} = \frac{1}{d^{1/p}}\norm{\cdot}_p$ is the normalized Schatten $p$-norm. We also need the power operator $I_{q,p}$ for $p,q \in \mathbb{R} \backslash \{0\}$: 
\begin{align*}
    I_{q,p}(X):= \Gamma_{\sigma}^{-1/q}\left(|\Gamma_{\sigma}^{1/p}(X)|^{p/q} \right)\,,\quad X \in \mc{B}(\mc{H}).
\end{align*} 
We summarize some important properties of $\norm{\cdot}_{p,\sigma}$ and $I_{p,q}$, see \cite[Lemmas 1,\,2]{kastoryano2013quantum} and \cite[Proposition 2.6]{gu2019interpolation}. 
\begin{lemma} \label{lem:prop:norm_power}
For $p,q,r \in \mathbb{R} \backslash \{0\}$ and $X \in \mathcal{B}(\mc{H})$, it holds that
\begin{enumerate}[1.]
\item $\norm{X}_{p,\sigma} \le \norm{X}_{q,\sigma}$ for $0 < p \le q$. 
    \item $\norm{I_{q,p}(X)}_{q,\sigma}^q = \norm{X}_{p,\sigma}^p$ and $I_{q,r} \circ I_{r,p} = I_{q,p}$. 
    \item $I_{p,p}(X) = X$ for $X \ge 0$. 
\end{enumerate}
\end{lemma}

We define the $p$-Dirichlet form ($p > 1$) for a QMS $\mc{P}_t$ with the generator $\mc{L}$  (our definition differs from the one in \cite{kastoryano2013quantum} by a factor of $p/2$):  for any full-rank invariant state $\sigma = \mc{P}_t^\dag(\sigma)$,  
\begin{align} \label{def:p_diri}
    \mc{E}_{p,\mc{L}}(X) := - \frac{\h{p}p}{4} \langle I_{\h{p},p}(X),\mc{L}(X) \rangle_{\sigma,1/2}\,,\quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
which, by Lemma \ref{lem:prop:norm_power}, readily gives  
\begin{align*}
    \mc{E}_{2,\mc{L}}(X) = - \langle X, \mc{L}(X) \rangle_{\sigma,1/2}\,, \quad X \ge 0\,.
\end{align*}
The $1$-Dirichlet form is defined as the limit $p \to 1^+$ of $\mc{E}_{p,\mc{L}}$ \cite[Proposition  8]{kastoryano2013quantum}:
\begin{align} \label{def:1diri}
  \mc{E}_{1, \mc{L}}(X) := \lim_{p \to 1^+} \mc{E}_{p,\mc{L}}(X) = - \frac{1}{4} \langle \log \Gamma_\si(X) - \log \sigma, \mc{L}(X) \rangle_{\sigma,1/2}\,,\quad X \ge 0\,.
\end{align}
For the generator $\mc{L}$ satisfying $\sigma$-DBC, by the formula \eqref{eq:inte_by_parts}, we have 
\begin{align} \label{eq:symm_p_diri}
      \mc{E}_{p,\mc{L}}(X) & = \frac{\h{p}p}{4} \sum_{j = 1}^J \langle \partial_j I_{\h{p},p}(X), \partial_j X \rangle_{\sigma,1/2}\,.
\end{align}
Thanks to the relation \eqref{eq:eigmodular}, we can further compute, for $p,q \neq 0$, 
\begin{align} \label{eq:partialpower_1}
    \partial_j I_{q,p}(X) & = V_j \Gamma_\sigma^{-1/q}\left(|\Gamma_\sigma^{1/p}(X)|^{p/q} \right) - \Gamma_\sigma^{-1/q}\left(|\Gamma_\sigma^{1/p}(X)|^{p/q} \right) V_j \notag \\
    & = \Gamma_\sigma^{-1/q} \left(  V_j |\Gamma_\sigma^{1/p}(e^{-\omega_j/2p}X)|^{p/q}  -  |\Gamma_\sigma^{1/p}(e^{\omega_j/2p}X)|^{p/q}  V_j \right)\,,
\end{align}
and, for $s \neq 0$,  
\begin{align} \label{eq:partialpower_2}
\partial_j X =  \Gamma_\sigma^{-1/s}\left(V_j \Gamma_\sigma^{1/s}(e^{-\omega_j/2s}X) - \Gamma_\sigma^{1/s}(e^{\omega_j/2s}X)V_j\right)\,.
\end{align}
Then, substituting the formulas \eqref{eq:partialpower_1} with $q=\h{p}$ and \eqref{eq:partialpower_2} with $s = p$ back into \eqref{eq:symm_p_diri}, and using Lemma \ref{lem:chain_rule}, we find the following representation of $\mc{E}_{p,\mc{L}}$: 
\begin{align} \label{eq:rep_epl}
        \mc{E}_{p,\mc{L}}(X)
       
        & = \frac{p^2}{4} \sum_{j = 1}^J \left\langle \Gamma_\si^{1/p} \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p}\Gamma_\si^{1/p}(X), e^{-\omega_j/2p}\Gamma_\si^{1/p}(X)\right)  \Gamma_\si^{1/p} \big(\partial_j X \big)  \right\rangle\,, \quad X \ge 0\,,
       
       
\end{align}
where $f_p^{[1]}(\cdot,\cdot)$ is the double sum operator associated with the divided difference of the function: 
\begin{equation} \label{def:funfp}
f_p(x): = \frac{1}{p-1} x^{p-1}\,,
\end{equation}
see \eqref{def:doutble_op_sum} and \eqref{def:divi_diff} for related definitions. 
Before we proceed, we derive some integral representation formulas of $f_p^{[1]}$ for later use.  We recall \cite[p.116]{bhatia2013matrix}
\begin{align*}
    x^{p - 1} = \frac{\sin(p \pi)}{\pi}\int_0^\infty \frac{ s^{p-1}}{s + x}\  ds \quad \text{for}\ x > 0\,,\ 0 < p < 1\,,
\end{align*}
which yields, for $1 < p < 2$,  
\begin{align} \label{eq:inte_divi_p}
f_p^{[1]}(x,y) & = \frac{1}{x - y} \int_x^y t^{p - 2}\ dt  \notag \\
  &  = \frac{1}{x - y} \frac{\sin((p-1) \pi)}{\pi} \int_x^y \int_0^\infty \frac{s^{p-2}}{s + t}\  d s  dt  \notag \\
  &  =  \frac{\sin((p - 1) \pi)}{\pi} \int_0^\infty s^{p-2} g_s^{[1]}(x,y)\ d s d t \,,
\end{align}
where $g_s(x) = \log(x + s)$ and $g_s^{[1]}$ is the associated divided differences. 
By the integral form of $g_0^{[1]}$:
\begin{align*}
g_0^{[1]}(x,y) = \int_0^\infty \frac{1}{(t + x)(t + y)} dt\,,
\end{align*}
we further have, from the formula \eqref{eq:inte_divi_p},  
\begin{align} \label{eq:integral_thetap}
    f_p^{[1]}(x,y)
  &  =  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \int_0^\infty \frac{1}{(t + s + x)(t + s + y)}\ dt ds\,.
\end{align}






We next recall the comparison result for the Dirichlet forms $\mc{E}_{p,\mc{L}}$, known as the quantum Stroock-Varopoulos inequality, that was proved in \cite[Theorem 14]{beigi2020quantum}. 
 \begin{lemma} \label{lem:qvs}
Let $\mc{L}$ be the generator of a QMS $\mc{P}_t$ that satisfies $\sigma$-DBC for some invariant state $\sigma \in \mc{D}_+(\mc{H})$, and $\mc{E}_{p,\mc{L}}$ be defined in \eqref{def:p_diri}. Then, for $X \ge 0$, we have 
\begin{equation} \label{eq:quantumsv}
    \mc{E}_{p,\mc{L}}(I_{p,2}(X)) \ge \mc{E}_{q,\mc{L}}(I_{q,2}(X))\,, \quad 0 < p \le q \le 2\,.
\end{equation}
\end{lemma}
In the special case $p \ge 1$ and $q = 2$, Lemma \ref{lem:qvs} gives the strong $L_p$ regularity of the Dirichlet form \cite{kastoryano2013quantum,bardet2017estimating}, which we slightly generalize as follows. The proof follows from the basic inequality: for $p \in (1,2]$, $a,b > 0$,
\begin{align*}
    (a-b)(a^{p-1} - b^{p-1}) \le \left( a^{p/2} - b^{p/2}\right)^2 \le \frac{p^2}{4(p-1)}(a-b)(a^{p-1} - b^{p-1})\,,
\end{align*}
and similar arguments in \cite[Theorem 4.1]{bardet2017estimating}. Hence we omit it here. 

\begin{corollary} \label{lem:com_diri}
Under the same assumptions as in Lemma \ref{lem:qvs}, it holds that, for $p \in (1,2]$ and $X \ge 0$, 
\begin{align} \label{eq:lpreg}
    \mc{E}_{2, \mc{L}}(I_{2,p}(X)) \le  \mc{E}_{p, \mc{L}}(X) \le \frac{p^2}{4(p-1)} \mc{E}_{2, \mc{L}}(I_{2,p}(X))\,.
\end{align}
In particular, the lower inequality in \eqref{eq:lpreg} holds for all $p \ge 1$. 
\end{corollary}

We now introduce the entropy function $\Ent_{p,\sigma}(X)$, for $p \ge 1$ and $\sigma \in \mc{D}_+(\mc{H})$, as follows \cite{beigi2020quantum} (our definition differs from the one in \cite{olkiewicz1999hypercontractivity,kastoryano2013quantum} by a factor of $p$):
\begin{align*}
    \Ent_{p,\sigma}(X) := \tr \left( \left(\Gamma_\si^{1/p}(X)\right)^p \left(\log \left(\Gamma_\si^{1/p}(X) \right)^p - \log \sigma \right) \right) - \norm{X}_{p,\sigma}^p \log \norm{X}_{p,\sigma}^p\,,\quad X \ge 0\,,
\end{align*}
and Umegaki's relative entropy:
\begin{equation} \label{def:relative_entropy}
    D(\rho \| \sigma) = \tr(\rho \log \rho - \rho \log \sigma)\,,\quad \rho \in \mathcal{D}(\mc{H})\,.
\end{equation}
We recall from \cite[Proposition 3]{beigi2020quantum} that
\begin{align} \label{eq:differ_norm}
\frac{d}{d p} \norm{Y}_{p,\sigma} = \frac{1}{p^2} \norm{Y}_{p,\sigma}^{1-p} 
\Ent_{p,\sigma}(I_{p,p}(Y))\,,\quad Y \in \mc{B}_{sa}(\mc{H})\,,
\end{align} 
which, by the chain rule, implies
\cite[Theorem 2.7]{olkiewicz1999hypercontractivity} 
\begin{equation} \label{eq:dev_pnorm}
    \frac{d}{d p} \norm{Y}_{p,\sigma}^p = \tr \Big((\Gamma_\sigma^{1/p}(Y))^p (\log \Gamma_\sigma^{1/p}(Y) - \frac{1}{p} \log \sigma ) \Big)\,,\quad Y \ge 0\,.
\end{equation}
The above formulas relate the differential of $L_p$-norm $\norm{\cdot}_{p,\sigma}$ and the entropy function $\Ent_{p,\sigma}$. The following lemma provides some basic properties of entropy functions $\Ent_{p,\sigma}(X)$; see \cite{kastoryano2013quantum,beigi2020quantum}. 

\begin{lemma} \label{lem:ent_rela}
For all $X \ge 0$, we have $\Ent_{p,\sigma}(X) \ge 0$. Moreover, for any density matrix $\rho \in \mc{D}(\mc{H})$, it holds that 
\begin{align*}
    \Ent_{2,\sigma}\big(\Gamma_\si^{-1/2}\left(\sqrt{\rho}\right)\big) = D(\rho || \sigma) \quad \text{and}\quad  \Ent_{1,\sigma}\big(\Gamma_\si^{-1}(\rho) \big) = D(\rho || \sigma)\,.
\end{align*}
\end{lemma}



We also recall the sandwiched R\'{e}nyi relative entropy 
introduced in \cite{muller2013quantum,wilde2014strong}: for $p \in (0,1) \cup (1,\infty)$,
\begin{align} \label{def:sandwi}
   D_p(\rho \| \sigma) := 
    \h{p}\log\big(\norm{\Gamma_\sigma^{-1}(\rho)}_{p,\sigma}\big), \quad  \rho \in \mathcal{D}(\mc{H})\,,
\end{align}
and 
\begin{align} \label{def:maxentropy}
    D(\rho\|\sigma) = \lim_{p \to 1} D_p(\rho\|\sigma)\,, \quad D_{\infty}(\rho \| \sigma) := \log \inf\{c > 0\,;\ \rho \le c \sigma
\} = \lim_{p \to \infty} D_p(\rho \| \sigma)\,,
\end{align}
where $ D(\rho\|\sigma)$ is the relative entropy \eqref{def:relative_entropy}, and $D_\infty(\rho\|\sigma)$ is the max-relative entropy \cite{datta2009min,muller2013quantum}.  It has been proved in \cite{frank2013monotonicity,beigi2013sandwiched} that $D_p$ satisfies the data processing inequality for $p \in (\frac{1}{2},1)\cup (1,\infty)$.
The operator $\Gamma_\sigma^{-1}(\rho)$ can be viewed as the relative density of a quantum state $\rho \in \mathcal{D}(\mc{H})$ with respect to the full-rank reference state $\sigma \in \mathcal{D}_+(\mc{H})$. For convenience, we shall call operators $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$ relative densities with respect to $\sigma$. In this work,  we mainly consider a modified sandwiched R\'{e}nyi entropy, referred to as \emph{quantum $p$-divergence}, 
\begin{align} \label{def:quanpdivi}
    \mc{F}_{p,\sigma}(\rho) := \frac{1}{p(p-1)} \big(\norm{\Gamma_\sigma^{-1}(\rho)}_{p,\sigma}^p - 1 \big)\,,\quad  \rho \in \mathcal{D}(\mc{H})\,,
\end{align}
where $\sigma \in \mathcal{D}_+(\mc{H})$ and $p \in (0,1) \cup (1,\infty)$. By definition, it follows that
\begin{equation} \label{eq:rela_sandpdivi}
     \mc{F}_{p,\sigma}(\rho) = \frac{1}{p(p-1)} \big(e^{(p-1)D_p(\rho\|\sigma)} - 1 \big)\,,
\end{equation}
and thus, by \eqref{def:maxentropy},
\begin{align*}
    \lim_{p \to 1} \mc{F}_{p,\sigma}(\rho)  = D(\rho\| \sigma)\,.
\end{align*}
We recall that the variance of $X \in \mc{B}_{sa}(\mc{H})$ is defined by 
\begin{align*}
    \Var_\sigma(X): = \norm{X - \tr(\sigma X)}_{2,\sigma}^2 = \norm{X}_{2,\sigma}^2 - \norm{X}_{1,\sigma}^2\,.
\end{align*}
Clearly, when $p = 2$, $\mc{F}_{p,\sigma}(\rho)$ reduces to the variance of the relative density of $\rho$, up to a constant factor,
\begin{align*}
    \mc{F}_{p,\sigma}(\rho) = \frac{1}{2} \Var_{\sigma}(X)\,, \quad X = \Gamma_\si^{-1} (\rho)\,.
\end{align*}
According to \eqref{eq:rela_sandpdivi}, many properties of $D_p(\rho\|\sigma)$ can be directly translated to $\mc{F}_{p,\sigma}(\rho)$ as follows. 
\begin{lemma} \label{lem:prop_divi}
For any $\rho \in \mathcal{D}(\mc{H}), \sigma \in \mathcal{D}_+(\mc{H})$, we have 
\begin{enumerate}[1.]
    \item $\mc{F}_{p,\sigma}(\rho) \ge 0$, and $ \mc{F}_{p,\sigma}(\rho) = 0$ if and only if $\rho = \sigma$. 
    \item $\mc{F}_{p,\sigma}(\rho)$ is jointly convex with respect to $(\rho,\sigma)$. 
    \item The data processing inequality holds for $ \mc{F}_{p,\sigma}(\rho)$,  $p \in (\frac{1}{2},1)\cup (1,\infty)$, 
    \begin{align} \label{eq:dpi_divi}
        \mc{F}_{p,\Phi(\sigma)}(\Phi(\rho)) \le \mc{F}_{p,\sigma}(\rho)\,,
    \end{align}
    where $\Phi$ is a quantum channel. 
\end{enumerate}
\end{lemma}










We finally recall the Araki–Lieb–Thirring (ALT) inequality \cite{lieb1997inequalities,araki1990inequality}, which is also useful in the sequel.
\begin{lemma}
For any $A \ge 0$, $B \ge 0$, and $q \ge 0$, it holds that 
\begin{align} \label{eq_alt_2}
    \tr( (B^r A^r B^r)^q) \le \tr( (BAB)^{rq})\,,\quad  0\le r \le 1\,. 
\end{align}
\end{lemma}




\section{Quantum interpolation functional inequalities} 
Recall that we limit our discussion to the primitive QMS $\mc{P}_t$ satisfying $\sigma$-DBC for some $\sigma \in \mathcal{D}_+(\mc{H})$. In this section, we introduce two new families of quantum functional inequalities: the quantum $p$-Beckner's inequality and the quantum dual $q$-Beckner's inequality, which interpolate the quantum Sobolev-type inequalities and the Poincar\'{e} inequality. To motivate the definition of quantum Beckner's inequalities, we consider the convergence of a QMS in terms of the quantum $p$-divergence $\mc{F}_{p,\sigma}$.  By the limit \eqref{eq:conver_qms} and the contractivity \eqref{eq:dpi_divi} of $\mc{F}_{p,\sigma}$ under CPTP maps, we have that $\mc{F}_{p,\sigma}$ decreases along the dynamic $\rho_t = \mc{P}_t^\dag(\rho)$ and there holds $\mc{F}_{p,\sigma}(\rho_t) \to 0$ as $t \to \infty$. Recalling that the convergence of the Umegaki’s relative entropy $D(\rho \| \sigma)$ is equivalent to the quantum modified log-Sobolev inequality  \cite{muller2016entropy,carlen2017gradient}, we will see that a similar result holds for $\mc{F}_{p,\sigma}(\rho)$, namely, the quantum Beckner's inequality characterizes the decay rate of $\mc{F}_{p,\sigma}(\rho_t)$. It is convenient to consider the dynamics of the relative density $X = \Gamma_\si^{-1}(\rho)$, equivalently, the QMS in the 
Heisenberg picture. From definition and Lemma \ref{lem:self_adjoint}, we have
\begin{align} \label{eq:evo_density}
    X_t := \Gamma_\si^{-1}(\rho_t) =  \Gamma_\si^{-1} e^{t \mc{L}^\dag} \Gamma_\si (X) = \mc{P}_t(X)\,.
\end{align}
Then we compute the time derivative of $\mc{F}_{p,\sigma}(\rho_t)$ as
\begin{align} \label{eq:ep_divi}
    \frac{d}{d t} \mc{F}_{p,\sigma}(\rho_t) & = \frac{1}{p-1} \langle \Gamma_\si^{-1/\h{p}}((\Gamma_\si^{-1/\h{p}}(\rho_t))^{p-1}), \mc{L}^\dag(\rho_t)\rangle \notag \\
    & = \frac{1}{p-1} \langle \Gamma_\si^{-1/\h{p}}(\Gamma_\si^{1/p}(X_t))^{p-1}),  \Gamma_\si\mc{L}(X_t)\rangle \notag \\ 
    & = - \frac{4}{p^2} \mc{E}_{p,\mc{L}}(X_t) \le 0\,.
\end{align}
Similarly to the case of $D(\rho \| \sigma)$ \cite{spohn1978entropy}, we can define the entropy production of the semigroup $\mc{P}_t$ associated with the $p$-divergence $\mc{F}_{p,\sigma}$ by the quantity $- \frac{4}{p^2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))$.  As a corollary of \eqref{eq:ep_divi}, we have the nonnegativity of the $p$-Dirichlet form:  $\mc{E}_{p,\mc{L}}(X) \ge 0$ for any $X \ge 0$. Moreover, again by \eqref{eq:ep_divi}, a simple use of Gr\"{o}nwall’s inequality concludes that the exponential decay of $\mc{F}_{p,\sigma}$:
\begin{align} \label{eq:exp_divi}
    \mc{F}_{p,\sigma}(\rho_t) \le e^{- \frac{4 \alpha_p}{p}t}  \mc{F}_{p,\sigma}(\rho)\,, \quad \forall\rho \in \mathcal{D}(\mc{H})\,,
\end{align}
for some $\alpha_p > 0$, is equivalent to the functional inequality:
\begin{equation} \label{eq_beck}
    \alpha_p \mc{F}_{p,\sigma}(\rho) \le p^{-1} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))\,, \quad \forall\rho \in \mathcal{D}(\mc{H})\,,
\end{equation}
which we refer to as the quantum Beckner's inequality. By analogy with the classical case \eqref{ineq_e}, the quantum dual Beckner's inequality can also be easily defined; see \eqref{ineq_dbecq} below. 







\subsection{Interpolation between Sobolev and Poincar\'{e} inequalities} \label{sec:inter_primi}

Let us now formally define the quantum functional inequalities in the Heisenberg picture that will be investigated in detail 
in the following sections. 

\begin{definition} \label{def:quantum_func} For a primitive QMS $\mc{P}_t$ with generator $\mc{L}$ satisfying $\sigma$-DBC, let $\mc{E}_{p,\mc{L}}$ be the $p$-Dirichlet form \eqref{def:p_diri}. Then we say that the QMS $\mc{P}_t$ satisfies:
\begin{enumerate}[1.]
    \item the Poincar\'{e} inequality if there exists a constant $\lambda > 0$ such that for all $X \in \mc{B}(\mc{H})$,
    \begin{equation} \label{ineq_pi}
        \lambda \norm{X - \tr(\sigma X) {\bf 1}}^2_{\sigma,f} \le - \langle X, \mc{L} X \rangle_{\sigma,f}\,, \tag{PI}
    \end{equation}
    where $f:(0,\infty) \to (0,\infty)$ and the norm $\norm{\cdot}_{\sigma,f}$ is defined by the inner product \eqref{eq:general_inner}. 
    \item  the modified log-Sobolev inequality (MLSI) if there exists $\alpha_1 > 0$ such that for all $X \ge 0$, 
\begin{equation} \label{ineq_mlsi}
    \alpha_1 \Ent_{1,\sigma}(X) \le \mc{E}_{1,\mc{L}}(X)\,. \tag{mLSI} 
\end{equation}
    \item the $p$-Beckner's inequality with $p \in (1,2]$ if there exists $\alpha_p > 0$ such that for all $X \ge 0$,
    \begin{align} \label{ineq_becp}
        \alpha_p (\norm{X}^p_{p,\sigma} - \norm{X}_{1,\sigma}^p) \le (p-1) \mc{E}_{p,\mc{L}}(X)\,. \tag{Bec-$p$}
    \end{align}
    \item the log-Sobolev inequality (LSI) if there exists  $\beta > 0$ such that for all $Y \ge 0$,  
    \begin{align} \label{ineq_lsi}
        \beta \Ent_{2,\sigma}(Y) \le \mc{E}_{2,\mc{L}}(Y)\,.  \tag{LSI}
    \end{align} 
    \item the dual $q$-Beckner's inequality with $q \in [1,2)$  if there exists  $\beta_q > 0$ such that for all $Y \ge 0$,
    \begin{align}  \label{ineq_dbecq}
        \beta_q \Var_{q,\sigma}(Y)\le (2-q) \mc{E}_{2,\mc{L}}(Y)\,,  \tag{Bec'-$q$}
    \end{align}
    where $\Var_{q,\sigma}(Y)$ is the $q$-variance: $\Var_{q,\sigma}(Y): = \norm{Y}_{2,\sigma}^2 - \norm{Y}_{q,\sigma}^2$.
\end{enumerate}
\end{definition}

The above functional inequalities can be easily reformulated in the Schr\"{o}dinger picture by inserting $X = \Gamma_\si^{-1}(\rho)$. For instance, from the definition of $\mc{F}_{p,\sigma}(\rho)$ \eqref{def:quanpdivi}, \eqref{ineq_becp} is clearly equivalent to \eqref{eq_beck}, while, by Lemma \ref{lem:ent_rela}, \eqref{ineq_mlsi} with $X = \Gamma_\si^{-1}(\rho)$ gives the familiar one in terms of quantum states \cite{rouze2019concentration}:
\begin{equation*}
    \alpha_1 D(\rho \| \sigma) \le - \frac{1}{4} \tr \left(\mc{L}^\dag(\rho) (\log \rho - \log \sigma) \right)\,, \quad \forall \rho \in \mathcal{D}(\mc{H})\,.
\end{equation*}
For ease of exposition, we call the optimal constant in \eqref{ineq_becp}
the quantum Beckner constant, denoted by $\alpha_p(\mc{L})$. Similar 
notions apply to other functional inequalities defined above. In particular, the Poincar\'{e} constant $\lambda(\mc{L})$ is nothing  but the spectral gap of $\mc{L}$, which is independent of the choice of $f$. Indeed, note from Lemma \ref{lem:self_adjoint} and Lemma \ref{lem:DBC} that $\mc{L}$ is self-adjoint with respect to $\langle\cdot,\cdot\rangle_{\sigma,f}$, and  $\ker({\mc{L}}) = \ker(\nabla) = {\rm span}\{ {\bf 1}_{\mc{H}} \}$ holds, which also equals to the commutant of operators $(V_j)_{j \in J}$. Then, by the min-max theorem, the Poincar\'{e} constant
\begin{equation}  \label{def:spect}
    \lambda(\mc{L}) := \inf_{X \in \mathcal{B}(\mc{H}), X \neq {\bf 1}} \frac{- \langle X, \mc{L} X \rangle_{\sigma,f}}{\norm{X - \tr(\sigma X){\bf 1}}^2_{\sigma,f}} 
\end{equation}
characterizes the smallest non-zero eigenvalue of $- \mc{L}$ (i.e., the spectral gap). It is worth pointing out that since the Lindbladian $\mc{L}$ is Hermitian-preserving, the infimum in \eqref{def:spect} can be equivalently taken 
over $X$ in $\mc{B}_{sa}(\mc{H})$.  




In this section, we discuss some properties of the optimal constants for the functional inequalities in Definition \ref{def:quantum_func} and the relations among them. The basic relations between \eqref{ineq_mlsi}, \eqref{ineq_lsi}, and \eqref{ineq_pi} have been  investigated and understood in \cite[Proposition 13 and Theorem 16]{kastoryano2013quantum}, which we state below for completeness and future use. 




 
\begin{lemma} \label{lem:aux_func}
For a primitive QMS $\mc{P}_t$ satisfying $\sigma$-DBC, it holds that
\begin{align*}
2 \beta(\mc{L}) \le  2 \alpha_1(\mc{L}) \le \lambda (\mc{L})\,.
\end{align*}
\end{lemma}

We first consider the properties of quantum dual $q$-Beckner's inequalities \eqref{ineq_dbecq}. It is clear from definition that when $q = 1$, the inequality \eqref{ineq_dbecq} reduces to the Poincar\'{e} inequality \eqref{ineq_pi} with $
f = x^{1/2}$. On the other hand, it is easy to see that in the limit $q \to 2$, \eqref{ineq_dbecq} gives the quantum LSI \eqref{ineq_lsi}. In this sense, \eqref{ineq_dbecq} can be considered as an interpolating family between the quantum LSI and the Poincar\'{e} inequality. Indeed, we have Proposition \ref{dualbecktolsi} below. The proof is based on the following monotonicity lemma. 


\begin{lemma}\label{lem:monotone}
The function $  \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2}$ is monotone increasing for $q \in [1,2)$.
\end{lemma}

\begin{proof}
We recall the interpolation of the noncommutative $L_p$ space
\cite[Corollary 3]{beigi2013sandwiched}: for $1\le p_0 < p_1 \le \infty$, and $1/p_\theta = (1-\theta)/p_0 + \theta/p_1$ with $\theta \in [0,1]$, 
$$
\norm{Y}_{p_\theta,\sigma} \le \norm{Y}_{p_0,\sigma}^\theta \norm{Y}_{p_1,\sigma}^{1-\theta}\,, \quad  \forall\, Y \in \mathcal{B}(\mc{H})\,.
$$ 
We immediately see that the function $\log \norm{Y}_{1/t,\sigma}$ is a convex function for $t \in [0,1]$, which implies that 
\begin{equation*}
\varphi(t) := \exp(2 \log \norm{Y}_{1/t,\sigma}) =\norm{Y}^2_{1/t,\sigma}    
\end{equation*}
is also convex. Therefore, we have that the function
\begin{align*}
    \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2} = \frac{\varphi(1/2) - \varphi(1/q)}{1/q - 1/2} 
\end{align*}
is increasing in $q$. 
\end{proof}


\begin{proposition} \label{dualbecktolsi} 
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS.  If \eqref{ineq_dbecq} holds for every $q \in [1,2)$ with $\limsup_{q \to 2^-}\beta_q > 0$, then \eqref{ineq_lsi} holds with constant $\beta \ge \limsup_{q \to 2^-} \beta_q/2$.  Conversely, if \eqref{ineq_lsi} holds with $\beta > 0$, then \eqref{ineq_dbecq} holds for every $q \in [1,2)$ with $\beta_q \ge q \beta$.
\end{proposition}

\begin{proof}
Suppose that \eqref{ineq_dbecq} holds with $\limsup_{q \to 2^-}\beta_q > 0$. By the formula \eqref{eq:differ_norm}, we have
\begin{align} \label{auxeq_1:limit}
    \frac{1}{2}\Ent_{2,\sigma}(Y) = \lim_{q \to 2^-}\frac{\norm{Y}_{2,\sigma}^2 - \norm{Y}_{q,\sigma}^2}{2-q}  \quad \text{for any}\ Y \ge 0\,.
\end{align}
Then taking the upper limit as $q \to 2^-$ in \eqref{ineq_dbecq}, we find 
\begin{align*}
   \frac{1}{2} \bigl(\limsup_{q \to 2^-} \beta_q\bigr) \Ent_{2,\sigma}(Y) \le \mc{E}_{2,\mc{L}}(Y)\,.
\end{align*}
Thus, by definition, \eqref{ineq_lsi} holds with $\beta \ge \limsup_{q \to 2^-} \beta_q/2$.  
For the reverse direction, by Lemma \ref{lem:monotone},
it follows, from \eqref{auxeq_1:limit} and the assumption that \eqref{ineq_lsi} holds, that 
\begin{align*}
    \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2} \le \limsup_{q \to 2^-}  \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2}  = 2 \Ent_{2,\sigma}(Y) \le 2\beta^{-1} \mc{E}_{2,\mc{L}}(Y)\,,
\end{align*}
that is, \eqref{ineq_dbecq} holds with $\beta_q \ge q \beta$.
\end{proof}

The quantum dual Beckner constant $\beta_q(\mc{L})$ has the following monotonicity property, which implies that if \eqref{ineq_dbecq} holds for some $q \in [1,2)$, then it holds for all $q \in [1,2)$. 


\begin{proposition} \label{prop:mono_dual_beck} 
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS.
For the optimal dual Beckner constant $\beta_q(\mc{L})$ in \eqref{ineq_dbecq}, it holds that $\beta_q(\mc{L})/(2-q)$ is increasing and  $\beta_q(\mc{L})/q$ is decreasing for $q \in [1,2)$. 
\end{proposition}

\begin{proof}
For the first claim, by the ordering of $\norm{\cdot}_{p,\sigma}$ in Lemma \ref{lem:prop:norm_power}, we have, for $ 1 \le q \le q' < 2$ and $Y \ge 0$,
\begin{equation*}
\beta_q(\norm{Y}^2_{2,\sigma} - \norm{Y}^2_{q',\sigma}) \le \beta_q(\norm{Y}^2_{2,\sigma} - \norm{Y}^2_{q,\sigma}) \le \frac{2-q}{2-q'} (2 - q') \mc{E}_{2,\mc{L}}(Y)\,,
\end{equation*}
that is, $\beta_{q'}/(2-q') \ge \beta_q/(2-q)$.  The second claim is a direct consequence of Lemma \ref{lem:monotone}. Indeed, due to the monotonicity, we have 
\begin{equation*}
    2q \beta_{q'} \frac{\Var_{q,\sigma}(Y)}{2 - q} \le 2q' \beta_{q'} \frac{\Var_{q',\sigma}(Y)}{2-q'} \le 2q'  \mc{E}_{2,\mc{L}}(Y)\,,
\end{equation*}
which clearly shows $\beta_q/q \ge \beta_{q'}/q'$. 
\end{proof}

We also prove an analog result for the quantum Beckner constant $\alpha_p(\mc{L})$.  


\begin{proposition} \label{prop:mono_beck} 
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC.
If the Beckner's inequality \eqref{ineq_becp} holds for some $p' \in (1,2]$ with $\alpha_{p'} > 0$, then for any $1 < p \le p'$, the inequality \eqref{ineq_becp} holds with constant $\alpha_p$ satisfying
\begin{align} \label{eq:monoconst}
    \frac{p}{p-1}\alpha_p \ge \frac{p'}{p'-1}\alpha_{p'}. 
\end{align}
Equivalently, $\frac{p}{p-1} \alpha_p(\mc{L})$, as a function of $p \in (1,2]$, is nonincreasing. 
\end{proposition}

\begin{proof}
It suffices to prove the inequality \eqref{eq:monoconst}.
For this, by the quantum Stroock-Varopoulos inequality in \eqref{eq:quantumsv} and \eqref{ineq_becp} for $p'$, we have
\begin{align} \label{auxeq:mono}
    \mc{E}_{p,\mc{L}}(X) \ge  \mc{E}_{p',\mc{L}}(I_{p',p}(X)) \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{I_{p',p}(X)}_{p',\sigma}^{p'} - \norm{I_{p',p}(X)}_{1,\sigma}^{p'}\right)\,.
\end{align}
Note from Lemma \ref{lem:prop:norm_power} that $\norm{I_{p',p}(X)}_{p',\sigma}^{p'} = \norm{X}_{p,\sigma}^p$. By ALT inequality \eqref{eq_alt_2}, we find
\begin{align*}
    \norm{I_{p',p}(X)}_{1,\sigma} & = \tr\left( \sigma^{\frac{1}{2\widehat{p'}}} \left(\sigma^{\frac{1}{2p}} X \sigma^{\frac{1}{2p}}\right)^{\frac{p}{p'}}\sigma^{\frac{1}{2\widehat{p'}}}\right) \le \norm{X}^{\frac{p}{p'}}_{\frac{p}{p'},\sigma}\,,
\end{align*}
since 
\begin{equation*}
    \frac{p}{p'} < 1\,,\quad \frac{1}{2\widehat{p'}} \frac{p'}{p} + \frac{1}{2p} = \frac{p'}{2p}\,.
\end{equation*}
Therefore, by \eqref{auxeq:mono} and Lemma \ref{lem:prop:norm_power}, it follows that 
\begin{equation*}
    \mc{E}_{p,\mc{L}}(X)  \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{X}_{p,\sigma}^p - \norm{X}^p_{\frac{p}{p'},\sigma}\right)  \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{X}_{p,\sigma}^p - \norm{X}^p_{1,\sigma}\right)\,.
\end{equation*}
The proof is complete by definition of Beckner's inequality. 
\end{proof}

We finally relate the $p$-Beckner's inequality and the dual $q$-Beckner's inequality with $q = 2/p$. 


\begin{proposition} \label{prop:beck_to_dual_beck} 
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC. 
Let $p \in (1,2]$ and $q = 2/p \in [1,2)$. If \eqref{ineq_dbecq} holds with $\beta_q$, then \eqref{ineq_becp} holds with $\alpha_p \ge p \beta_q/2$.
\end{proposition}

\begin{proof}
We substitute $Y  = I_{2,p}(X)$ for $X \ge 0$ in \eqref{ineq_dbecq} and find 
\begin{equation} \label{auxeq_dualbeck_to_beck}
    \norm{I_{2,p}(X)}^2_{2,\sigma} - \norm{I_{2,p}(X)}_{2/p,\sigma}^{2} = \norm{X}^p_{p,\sigma} - \norm{I_{2,p}(X)}_{2/p,\sigma}^{2} \le \beta_q^{-1}(2-q) \mc{E}_{2,\mc{L}}(I_{2,p}(X))\,.
\end{equation}
By ALT inequality \eqref{eq_alt_2} and Lemma \ref{lem:prop:norm_power}, we have 
\begin{align} \label{auxeq_dualbeck_to_beck_2}
    \norm{I_{2,p}(X)}_{2/p,\sigma}^{2/p} & = \tr \big(\sigma^{\frac{p}{4} - \frac{1}{4}} \big(\sigma^{\frac{1}{2p}} X \sigma^{\frac{1}{2p}}  \big)^{\frac{p}{2}}\sigma^{\frac{p}{4} - \frac{1}{4}}\big)^{\frac{2}{p}} \notag \\
    & \le \tr \bigl((\Gamma_\sigma(X))^{\frac{p}{2}}\bigr)^{\frac{2}{p}} \le \norm{X}_{1,\sigma}.
\end{align}
Then, by $L_p$ regularity in Corollary \ref{lem:com_diri}, it follows from \eqref{auxeq_dualbeck_to_beck} and \eqref{auxeq_dualbeck_to_beck_2} that 
\begin{align*}
  \norm{X}^p_{p,\sigma} - \norm{X}_{1,\sigma}^p \le \beta_q^{-1}(2-q) \mc{E}_{p,\mc{L}}(X)\,,
 \end{align*}
 which gives $\alpha_p \ge p \beta_q/2$. 
\end{proof}


\subsection{Quantum Beckner constant} \label{sec:positive_stab}
In this section, we focus on the properties of the quantum Beckner constant $\alpha_p(\mc{L})$. We first provide a two-sided bound for $\alpha_p(\mc{L})$ in terms of the Poincar\'{e} constant $\lambda(\mc{L})$, which will yield the uniform positivity of $\alpha_p(\mc{L})$ (i.e., $\inf_{p \in (1,2]}\alpha_p(\mc{L}) > 0$). Then, we investigate the relations between the quantum MLSI \eqref{ineq_mlsi} and the $p$-Beckner’s inequality \eqref{ineq_becp}. Finally, we will extend the quantum  Holley-Stroock’s argument from \cite{junge2019stability} and give the stability estimates for the Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state $\sigma$. 


\subsubsection{Uniform positivity of $\alpha_p(\mc{L})$}

Let us start with the following proposition generalizing \cite[Proposition 2.8]{adamczak2022modified} in the classical setting to the quantum regime, which shows that the quantum Poincar\'{e} inequality \eqref{ineq_pi} implies $p$-Beckner's inequality \eqref{ineq_becp} for $p \in (1,2]$. 

\begin{proposition} \label{prop:pb} 
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC. 
If the Poincar\'{e} inequality \eqref{ineq_pi} holds with constant $\lambda$, then, for all $p \in (1,2]$, the Beckner's inequality \eqref{ineq_becp} holds with constant $\alpha_p$ satisfying 
\begin{equation} \label{eq:poin_to_beck}
\alpha_p \ge \lambda (p-1)\,.
\end{equation}
In particular, if \eqref{ineq_mlsi} holds  with constant $\alpha_1$, then $\alpha_p \ge 2 \alpha_1(p-1)$. 
\end{proposition}

\begin{proof}
We first claim that there holds 
\begin{equation} \label{auxeq:convex_bec}
      \norm{X}_{p,\sigma}^p -  \norm{X}_{1,\sigma}^p \le \langle I_{\h{p},p}(X), X - \tr(\sigma X) {\bf 1} \rangle_{\sigma,1/2}\,,\quad \forall X \ge 0\,.
\end{equation}
Indeed, a direct computation gives 
\begin{align} \label{auxeq:ipp}
     \left\langle I_{\h{p},p}(X), X - \tr(\sigma X) {\bf 1} \right\rangle_{\sigma,1/2} = \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma} \tr(\sigma^{1/p}(\Gamma_\si^{1/p}X)^{p-1})\,. 
\end{align}
Then, by ALT inequality \eqref{eq_alt_2} and the ordering of $\norm{\cdot}_{p,\sigma}$ in Lemma \ref{lem:prop:norm_power}, we have
\begin{align*}
   \tr(\Gamma_\si^{1/p}(\Gamma_\si^{1/p}X)^{p-1}) & \le  \tr((\Gamma_\si^{1/(p-1)}X )^{p-1}) = \norm{X}_{p-1,\sigma}^{p-1} \le \norm{X}_{1,\sigma}^{p-1}\,,
\end{align*}
which, along with \eqref{auxeq:ipp},  implies the desired inequality \eqref{auxeq:convex_bec}. 
One can readily note that up to some constant, the right hand term in \eqref{auxeq:convex_bec} is the $p$-Dirichlet form $\mc{E}_{p,\mc{L}_{depol}}$ defined by \eqref{def:p_diri} 
associated with the generator $\mc{L}_{depol}$ with $\gamma = 1$; see Example \ref{exp:dep_channel}. We proceed by using Corollary \ref{lem:com_diri} and find 
\begin{align} \label{auxeq:diri_depo}
     \mc{E}_{p, \mc{L}_{depol}}(X) \le \frac{p^2}{4(p-1)} \Var_{2,\sigma}(I_{2,p}(X)) \le \frac{1}{\lambda}\frac{p^2}{4(p-1)} \mc{E}_{2,\mc{L}}(I_{2,p}(X)) \le \frac{1}{\lambda}\frac{p^2}{4(p-1)} \mc{E}_{p, \mc{L}}(X)\,,
\end{align}
where we have also used the assumption that \eqref{ineq_pi} holds, and the observation:
\begin{equation*}
    \Var_{2,\sigma}(X) = \langle X - \tr(\sigma X) {\bf 1}, X - \tr(\sigma X) {\bf 1} \rangle_{\sigma,1/2} =  \mc{E}_{2,\mc{L}_{depol}}(X)\,.
\end{equation*}
Then it follows from \eqref{auxeq:convex_bec} and \eqref{auxeq:diri_depo} that 
\begin{align*}
\lambda (\norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^p) \le   \mc{E}_{p, \mc{L}}(X)\,,
\end{align*}
that is, $\alpha_p \ge \lambda (p-1)$ holds. The proof is completed by Lemma \ref{lem:aux_func}. 
\end{proof}

Unfortunately, the lower bound for the Beckner constant $\alpha_p(\mc{L})$ provided in \eqref{eq:poin_to_beck} 
vanishes as $p \to 1^+$, and the proof techniques used in the classical case \cite[Proposition 2.9 and Theorem 2.1]{adamczak2022modified} to deal with this limiting regime cannot be easily extended to the noncommutative setting. 
In the following, we establish a uniform lower bound for $\alpha_p(\mc{L})$ by adopting the analysis framework recently proposed in \cite{gao2021complete} for the complete modified log-Sobolev inequality. We first extend the key estimates in \cite[Lemma 2.1 and Lemma 2.2]{gao2021complete} to our case. 


\begin{lemma}\label{lem:mono_norm}
Let operators $X_i, Y_i > 0$, $ i = 1,2$, satisfy $X_i \le c Y_i$ for some $c > 0$. It holds that 
\begin{align} \label{eq:mono_norm_two}
\langle A, f_p^{[1]}(Y_1, Y_2) A\rangle \le c^{2-p} \langle A , f_p^{[1]}( X_1,  X_2)  A \rangle\,, \quad \forall A \in \mathcal{B}(\mc{H})\,,
\end{align}
where $f_p^{[1]}$ is the divided difference of the function $f_p$ defined in \eqref{def:funfp}. 
\end{lemma}

\begin{proof}
The proof follows from the integral form \eqref{eq:integral_thetap} of $f_p^{[1]}$: 
\begin{align} \label{eq:auxest_mono} 
 \langle A,  f_p^{[1]}(X_1, X_2) A\rangle
  &  =  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \int_0^\infty \tr \left( A^* \frac{1}{t + s + X_1} A \frac{1}{t + s + X_2} \right) dt ds \notag \\
 & \ge  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \int_0^\infty \tr \left(A^* \frac{1}{t + s +  c Y_1} A \frac{1}{t + s + c Y_2} \right) dt ds \notag \\
  & =  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \frac{1}{c^2} \int_0^\infty \tr \left(A^* \frac{1}{\frac{t + s}{c} +  Y_1} A \frac{1}{\frac{t + s}{c} +  Y_2} \right) dt ds \notag \\
   & =  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  (cs)^{p-2} \int_0^\infty \tr \left(A^* \frac{1}{t + s + Y_1} A \frac{1}{t + s + Y_2} \right) dt ds \\
   & = c^{p-2}  \langle A,  f_p^{[1]}(Y_1, Y_2) A \rangle\,,\notag
\end{align}
where in the first inequality we have used the fact that $t^{-1}$ is operator monotone decreasing; in the third line we have used the change of variable $r \to r/s$ and $t \to t/s$. 
\end{proof}



For our subsequent discussion, several useful observations are in order. First, we define the function
\begin{align*}
    \varphi_p(x) = \frac{1}{p-1}\frac{x - x^{1/p}}{x^{1/p} - 1}\,.
\end{align*}
Then the kernel $J_\sigma^{\varphi_p} = \varphi_p(\Delta_\sigma) R_\sigma $ for the inner product $\langle \cdot, \cdot\rangle_{\sigma,\varphi_p}$ (cf.\,\eqref{def:operator_kernel} and \eqref{eq:general_inner}) can be reformulated, in terms of the double sum operator associated with $f_p^{[1]}$, as follows: 
\begin{align} \label{eq:kernel_iden}
   \varphi_p(\Delta_\sigma) R_\sigma & = \frac{1}{p-1} \frac{\Delta_{\sigma} - \Delta_\sigma^{1/p}}{\Delta_{\sigma}^{1/p} - 1} R_\sigma \notag \\
   & = \frac{1}{p-1} \frac{L_{\sigma}^{(p-1)/p} - R^{(p-1)/p}_\sigma}{L_{\sigma}^{1/p} - R^{1/p}_\sigma}L_\sigma^{1/p}R_\sigma^{1/p} \notag \\ & = \Gamma_\si^{1/p} f_p^{[1]}\left(\sigma^{1/p},\sigma^{1/p}\right)\Gamma_\si^{1/p}\,.
\end{align}
Second, it turns out that the inner product $\langle \cdot , \cdot  \rangle_{\sigma,\varphi_p}$ is also related to the $\chi^2$-divergence associated with the power difference $\kappa_\alpha$ (cf.\,Definition \ref{def:quantumchi} and \eqref{def:pdiff}). Indeed, for $\rho \in \mathcal{D}(\mc{H})$ with the relative density $X = \Gamma_\si^{-1}(\rho)$, we can directly compute 
\begin{equation*}
  \norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 = \langle \rho - \sigma, \Gamma_\si^{-1}\varphi_p(\Delta_\sigma)R_\sigma \Gamma_\si^{-1}(\rho - \sigma)\rangle = \langle \rho-\sigma, \Omega_\sigma^{x^{-1}\varphi_p}(\rho - \sigma)\rangle\,,
\end{equation*}
by noting
\begin{align}  \label{eq:kernel_iden_2}
    \Gamma_\si^{-1}\varphi_p(\Delta_\sigma)R_\sigma \Gamma_\si^{-1} = \Delta_\sigma^{-1}\varphi_p(\Delta_\sigma)R_\sigma^{-1} = \Omega_\sigma^{x^{-1}\varphi_p}\,,
\end{align}
see \eqref{def:omega} for the definition of the operator $\Omega_\sigma^{x^{-1}\varphi_p}$.  Then we find, by setting $\alpha = 1/p$ and \eqref{def:pdiff}, 
\begin{align} \label{auxeq_power_diff}
    x^{-1}\varphi_{1/\alpha}(x) = \frac{1}{p-1}\frac{1-x^{(1-p)/p}}{x^{1/p}-1} = \frac{\alpha}{\alpha-1} \frac{x^{\alpha-1} - 1}{x^{\alpha} - 1} = \kappa_\alpha\,.
\end{align}
With the notation defined above, we can conclude
\begin{align} \label{eq:chi_heisen}
    \norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 = \chi^2_{\kappa_\alpha}(\rho,\sigma)\,.
\end{align}
Third, by Lemma \ref{lem:DBC}, it holds that
\begin{align} \label{eq:rep_dbcl}
   - \left\langle X, \mc{L} X \right\rangle_{\sigma,\varphi_p}
   & = \left\langle \partial_j X,  R_{e^{-\omega_j/2} \sigma} \varphi_p\left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2} \sigma}\right) \partial_j X \right\rangle
   \notag \\
   & =  \left\langle \Gamma_\si^{1/p}  \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p}\sigma^{1/p}, e^{-\omega_j/2p}\sigma^{1/p}\right) \Gamma_\si^{1/p}\big(\partial_j X\big) \right\rangle\,,\quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
where the second equality follows from the same calculation as in \eqref{eq:kernel_iden}. 


We are now ready to give the following crucial lemma that provides a two-sided estimate of the quantum $p$-divergence $\mc{F}_{p,\sigma}(\rho)$ in terms of the quantum $\chi^2$-divergence associated with the power difference $\kappa_{1/p}$.

 

\begin{lemma} \label{lem:two-sided}
For $\rho \in \mathcal{D}_+(\mc{H})$ satisfying $\rho \le c \sigma$ for some $c > 0$, it holds that
\begin{equation} \label{eq:two_sided_dense}
    k_p(c) \chi_{\kappa_{1/p}}^2(\rho,\sigma) \le \mc{F}_{p,\sigma}(\rho) \le p^{-1} \chi_{\kappa_{1/p}}^2(\rho,\sigma)\,,
\end{equation}
where the constant $k_p(c)$ is given by 
\begin{align*}
    k_p(c) =  \frac{c^p - 1 - p(c-1)}{p(c-1)^2 (p-1)}. 
\end{align*}
In particular, the upper bound estimate in \eqref{eq:two_sided_dense} holds for any $\rho \in \mathcal{D}_+(\mc{H})$. 
\end{lemma}


\begin{proof}
Recalling the relation \eqref{eq:chi_heisen}, we will prove the inequality \eqref{eq:two_sided_dense} in the Heisenberg picture:
\begin{align} \label{eq:two_sided_est}
 k_p(c) \norm{X - {\bf 1}}^2_{\sigma,\varphi_p}   \le  \frac{1}{p(p-1)}\left(\norm{X}_{p,\sigma}^p - 1\right) \le p^{-1} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p},
\end{align}
for $X > 0$ satisfying $\tr(\sigma X) = 1$ and $X \le c {\bf 1}$. 
We define $X_t = (1 - t) {\bf 1} + t X$, $t \in [0,1]$, and consider the function: 
\begin{equation*}
    \varphi(t) := \frac{1}{p(p-1)}\left(\norm{X_t}_{p,\sigma}^p - 1 \right). 
\end{equation*}
It is easy to compute the derivatives:
\begin{align*}
    \varphi'(t) = \frac{1}{p-1}\tr \left( \left( \Gamma_\si^{1/p}(X_t)\right)^{p-1} \Gamma_\si^{1/p}\left(X-{\bf 1}\right) \right)\,,
\end{align*}
and 
\begin{align} \label{auxeq:2nd_der}
    \varphi''(t) = \left\langle \Gamma_\si^{1/p}(X-{\bf 1}), f_p^{[1]}\left(\Gamma_\si^{1/p}(X_t),\Gamma_\si^{1/p}(X_t) \right) \Gamma_\si^{1/p}(X-{\bf 1}) \right\rangle. 
\end{align}
By assumption $X \le c {\bf 1}$, we have
\begin{align} \label{aauxeq}
(1 - t) {\bf 1}   \le X_t \le (1 + (c - 1) t) {\bf 1}\,,
\end{align}
and hence $(1-t) \sigma^{1/p}\le \Gamma_\si^{1/p}(X_t) \le (1 + (c - 1) t) \sigma^{1/p}$. 
Then applying Lemma \ref{lem:mono_norm} to \eqref{auxeq:2nd_der} gives 
\begin{align*}
(1 + (c-1)t)^{p-2} \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle  \le \varphi''(t) \le  (1 - t)^{p-2} \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle,
\end{align*}
where we have used \eqref{aauxeq} and the observation \eqref{eq:kernel_iden}. 
Note from $\tr (\sigma X) = 1$ that $\varphi'(0)= 0$. It follows that 
\begin{align*}
    \varphi(1) - \varphi(0) = \int_0^1 \int_0^x \varphi''(t) dt dx & \le   \int_0^1 \int_0^x  (1 - t)^{p-2} dt dx \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle \\
    & \le  p^{-1} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p}\,.
\end{align*}
Similarly, for the lower bound, we have 
\begin{align*}
    \varphi(1) - \varphi(0) & \ge   \int_0^1 \int_0^x  (1 + (c - 1) t)^{p-2} dt dx \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle \\
    & \ge \frac{c^p - 1 - p(c-1)}{p(c-1)^2 (p-1)} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p}. 
\end{align*}
The proof is complete by noting $\varphi(0) = 0$. 
\end{proof}

With the help of above lemmas, we can extend \cite[Theorem 3.3]{gao2021complete} to  uniformly bound the Beckner constant $\alpha_p(\mc{L})$. For this, we define a constant for the invariant state $\sigma$ of $\mc{P}_t^\dag$:
\begin{align} \label{def:constcsi}
    C(\sigma):= & \inf\{c > 0\,;\ \rho \le c \sigma
 \ \text{for all}\ \rho \in \mc{D}(\mc{H})
\}\,.
\end{align}
It is easy to note that $C(\sigma)$ is closely related to the max-relative entropy $D_{\infty}$ in \eqref{def:maxentropy} and can be explicitly represented:
\begin{align*} 
    C(\sigma) = \sup_{\rho \in \mc{D}(\mc{H})} \exp\left(D_{\infty}(\rho \| \sigma) \right) = \sigma_{\min}^{-1}\,,
\end{align*}
where $\sigma_{\min}$ is the smallest eigenvalue of $\sigma$. Indeed, \cite[Lemma 2.1]{muller2018sandwiched} shows that for $\sigma \in \mathcal{D}_+(\mc{H})$ and $p \in [1,\infty]$, the sandwiched R\'{e}nyi relative entropy $D_p$ satisfies
\begin{align} \label{eq:suprhodp}
\sup_{\rho \in \mathcal{D}(\mc{H})} D_p(\rho \| \sigma) = \log \sigma_{\min}^{-1}\,.
\end{align}
We remark here that $\sigma_{\min}$ has been  estimated in several interesting cases; see \cite[Remark 1]{kastoryano2013quantum} and \cite[Section 7]{muller2018sandwiched}. 






\begin{theorem} \label{thm:beck_poincare}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS satisfying $\sigma$-DBC. Then the Beckner's inequality  \eqref{ineq_becp} holds for all $p \in (1,2]$ with constant $\alpha_p(\mc{L})$ satisfying the estimate:
\begin{align} \label{eq:beck_poin}
    \frac{p^2 \sigma_{\min}^{2-p}}{4} \lambda(\mc{L}) \le  \alpha_p(\mc{L}) \le \frac{p}{2}\lambda(\mc{L})\,,
\end{align}
where $\lambda(\mc{L})$ is the Poincar\'{e} constant.
\end{theorem}

\begin{proof} 
We first show the right inequality in \eqref{eq:beck_poin}, that is, \eqref{ineq_becp} implies \eqref{ineq_pi}, by the standard linearization argument.
We consider $Z = {\bf 1} + \varepsilon X$ for an arbitrary $X \in \mc{B}_{sa}(\mc{H})$ with $\tr(\sigma X) = 0$, where $\varepsilon > 0$ is small enough such that $Z > 0$. By a direct expansion with respect to $\varepsilon$, we find 
\begin{equation} \label{eq:asy_divi}
    \norm{Z}_{p,\sigma}^p = 1 + \varepsilon p \tr(\sigma X) + \frac{\varepsilon^2}{2}p(p-1)\norm{X}_{\sigma,\varphi_p}^2 + O(\varepsilon^3)\,,
\end{equation}
and 
\begin{align*}
    \norm{Z}_{1,\sigma}^p = \big(1 + \varepsilon \tr(\sigma X)\big)^p = 1 + \varepsilon p \tr(\sigma X) + \frac{\varepsilon^2}{2}p(p-1)\tr(\sigma X)^2 + O(\varepsilon^3)\,.
\end{align*}
We also compute 
\begin{align*}
 \Gamma_\si I_{\h{p},p}(Z) = \sigma + \varepsilon (p-1) \varphi_p(\Delta_\sigma)R_\sigma(X) + O(\varepsilon^2)\,,
\end{align*}
and $\mc{L}(Z) = \varepsilon \mc{L}(X)$, which yields 
\begin{align} \label{eq:asy_diri}
    \mc{E}_{p,\mc{L}}(Z) = -  \varepsilon^2(p-1)\frac{\h{p}p}{4} \langle X, \mc{L} X\rangle_{\sigma,\varphi_p} + O(\varepsilon^3)\,. 
\end{align}
Hence, by applying \eqref{ineq_becp} to $Z$, we have
\begin{align} \label{midauxeq}
    \alpha_p \frac{\varepsilon^2}{2}p(p-1) \left(\norm{X}_{\sigma,\varphi_p}^2 - \tr(\sigma X)^2\right) + O(\varepsilon^3) 
    \le -  \varepsilon^2(p-1)^2\frac{\h{p}p}{4} \langle X, \mc{L} X\rangle_{\sigma,\varphi_p} + O(\varepsilon^3)\,.
\end{align}
Due to $\tr(\sigma X) = 0$, it readily follows that $$2 \alpha_p \norm{X}_{\sigma,\varphi_p}^2 \le - p \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,,$$ by dividing both sides of \eqref{midauxeq} by $\varepsilon^2$ and letting $\varepsilon \to 0$, 
which gives the desired estimate: $2 \alpha_p \le p \lambda(\mc{L})$. 


We next show the left inequality in \eqref{eq:beck_poin}. 
It suffices to consider relative densities $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$, for which there holds $X \le C(\sigma) {\bf 1} $, by definition \eqref{def:constcsi} of the constant $C(\sigma)$.  Using the upper estimate in \eqref{eq:two_sided_est} and the positivity of the spectral gap of $\mc{L}$,  we have 
\begin{align} \label{auxeqq_1}
    \frac{1}{p-1}(\norm{X}_{p,\sigma}^p - 1) \le  \norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 \le - \lambda(\mc{L})^{-1} 
    \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,.
   
   
\end{align}
Then, by the formulas \eqref{eq:rep_epl}, \eqref{eq:rep_dbcl}, and Lemma \ref{lem:mono_norm}, it follows that
\begin{align} \label{auxeqq_2}
 - \left\langle X, \mc{L} X \right\rangle_{\sigma,\varphi_p}
   & =  \left\langle \Gamma_\si^{1/p} \big(\partial_j X \big), f_p^{[1]}\left(e^{\omega_j/2p}\sigma^{1/p}, e^{-\omega_j/2p}\sigma^{1/p}\right) \Gamma_\si^{1/p} \big(\partial_j X\big) \right\rangle  \notag \\
   & \le C(\sigma)^{2-p} \left\langle \Gamma_\si^{1/p}  \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p} \Gamma_\si^{1/p}(X), e^{-\omega_j/2p} \Gamma_\si^{1/p}(X) \right) \Gamma_\si^{1/p}\big(\partial_j X\big) \right\rangle \notag \\
   & \le 4 p^{-2} C(\sigma)^{2-p} \mc{E}_{p,\mc{L}}(X)\,.
\end{align}
Therefore, by \eqref{auxeqq_1} and \eqref{auxeqq_2}, we obtain
\begin{equation*}
    \frac{1}{p-1}(\norm{X}_{p,\sigma}^p - 1) \le \frac{4 C(\sigma)^{2-p}}{p^2 \lambda(\mc{L})} 
    \mc{E}_{p,\mc{L}}(X)\,. \qedhere
\end{equation*}
\end{proof}




For a specific invariant state $\sigma$, the lower bound $\alpha_p \ge \lambda (p-1)$ in \eqref{eq:poin_to_beck} may be better than the one in \eqref{eq:beck_poin} for some values of $p \in (1,2]$. Combining these two estimates, we obtain a sharper lower bound for $\alpha_p$. 



\begin{corollary} \label{coro:pitobeck}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS satisfying $\sigma$-DBC. Then the Beckner constant $\alpha_p(\mc{L})$ satisfies
\begin{equation} \label{eq:lower_est}
\alpha_p(\mc{L}) \ge \max\Big\{ \lambda(\mc{L}) (p-1)\,,  \frac{p^2 \sigma_{\min}^{2-p}}{4} \lambda(\mc{L}) \Big\}\,.
\end{equation}
The lower bound \eqref{eq:lower_est} still holds if we replace $\lambda(\mc{L})$ by $2 \alpha_1(\mc{L})$. 
\end{corollary}

\subsubsection{Connections with $\alpha_1(\mc{L})$}
 We next discuss the relations between the quantum $p$-Beckner's inequalities \eqref{ineq_becp} and the quantum MLSI \eqref{ineq_mlsi}. 
First, by dividing both sides of \eqref{ineq_becp} by a factor $(p-1)$, and then taking the right limit $p \to 1^+$ and using formulas \eqref{def:1diri} and \eqref{eq:dev_pnorm}, we obtain the following lemma. 

\begin{lemma} \label{lem:mb} 
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS. 
If \eqref{ineq_becp} holds for all $p \in (1,2]$ with $\limsup_{p \to 1^+} \alpha_p > 0$, then \eqref{ineq_mlsi} holds with 
\begin{equation} \label{eq:beckmsli_one}
    \alpha_1 \ge \limsup_{p \to 1^+} \alpha_p\,.
\end{equation}
\end{lemma}


\begin{theorem} \label{thm:beck_log_sobo}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS with $\sigma$-DBC. Then for every $p \in (1,2]$, the Beckner's inequality \eqref{ineq_becp} holds 
if and only if the modified log-Sobolev inequality \eqref{ineq_mlsi} holds. Moreover, there holds 
\begin{equation} \label{eq:opt_becktoopt_msli}
    \alpha_1(\mc{L}) = \lim_{p \to 1^+} \alpha_p({\mc{L}})\,.
\end{equation}
\end{theorem}

The proof of \eqref{eq:opt_becktoopt_msli} depends on the following lemma that extends the result \cite[Theorem 6.5]{bobkov2006modified} for the discrete modified log-Sobolev inequality.  

\begin{lemma} \label{lem:extremal_func} 
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS. 
If $\alpha_p(\mc{L}) < p \lambda(\mc{L})/2$ holds for the quantum Beckner constant, then the following infimum is attained:
\begin{align} \label{auxeq:defoptap}
    \alpha_{p}(\mc{L}) = \inf_{\substack{X\ge 0, X \neq {\bf 1} \\ \norm{X}_{1,\sigma} = 1}} \frac{ (p-1)\mc{E}_{p,\mc{L}}(X)}{\norm{X}_{p,\sigma}^p- 1}\,.
\end{align} 
\end{lemma}

\begin{proof}
By definition, there exists a sequence of $X_n \in \{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\,,\, X \neq {\bf 1}\}$ such that 
\begin{align} \label{eq:assp}
    V_p(X_n): = \frac{ (p-1)\mc{E}_{p,\mc{L}}(X_n)}{\norm{X_n}_{p,\sigma}^p- 1} \to \alpha_p(\mc{L})\,,\quad \text{as}\ n \to \infty\,.
\end{align}
Since the set $\{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\}$ is compact, without loss of generality, we assume that $X_n \to X $ as $n \to \infty$ for some $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$. Suppose that $X = {\bf 1}$. Then we can write $X_n = {\bf 1} + Y_n$ with $Y_n \to 0$ and $\tr(\sigma Y_n) = 0$. Recalling the asymptotic expansions \eqref{eq:asy_divi} and \eqref{eq:asy_diri}, we obtain, by \eqref{ineq_pi}, 
\begin{align*}
\liminf_{n \to \infty} V_p(X_n) = \liminf_{n \to \infty} - \frac{p}{2} \frac{\langle Y_n, \mc{L} Y_n \rangle_{\sigma,\varphi_p} + O(\norm{Y_n}^3_{1,\sigma})}{\norm{Y_n}^2_{\sigma,\varphi_p} + O(\norm{Y_n}^3_{1,\sigma})} \ge \frac{p}{2}\lambda(\mc{L}) > \alpha_p(\mc{L})\,,
\end{align*}
which contradicts \eqref{eq:assp}. Thus, the operator $X$ is in the desired set $\{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\,,\, X \neq {\bf 1}\}$ and the infimum in \eqref{auxeq:defoptap} is attained. 
\end{proof}




\begin{proof}[Proof of Theorem \ref{thm:beck_log_sobo}]
The first statement follows from Theorem \ref{thm:beck_poincare} and \cite[ Theorem 3.3]{gao2021complete}. 
In order to establish \eqref{eq:opt_becktoopt_msli}, by Lemma \ref{lem:mb}, it suffices to prove 
\begin{align} \label{auxeq:infmb}
    \liminf_{p \to 1^+}  \alpha_p (\mc{L}) \ge  \alpha_1 (\mc{L})\,.
\end{align}
We shall prove it by contradiction. If \eqref{auxeq:infmb} does not hold, there is a sequence $p_n \to 1^+$ as $n \to \infty$ such that 
\begin{align*} 
    \lim_{n \to \infty}  \alpha_{p_n}(\mc{L}) \le \alpha_1(\mc{L}) - \varepsilon\,, 
\end{align*}
for some small enough $\varepsilon > 0$, that is, for any $k > 0$, there exists $N$ such that for $n \ge N$, 
\begin{align} \label{auxeq:seq}
   \alpha_{p_n}(\mc{L}) \le \alpha_1(\mc{L}) - \varepsilon + \frac{1}{k}\,.
\end{align}
Suppose that $\alpha_{p_n}(\mc{L}) = {p_n}\lambda(\mc{L})/2$ holds for infinitely many $n$. It follows from \eqref{auxeq:seq} that $\lambda(\mc{L})/2 \le \alpha_1(\mc{L}) - \varepsilon$, which is a contradiction with Lemma \ref{lem:aux_func}. Hence, without loss of generality, we assume that $\alpha_{p_n}(\mc{L}) < {p_n}\lambda(\mc{L})/2$ for all $n$. Then, Lemma \ref{lem:extremal_func} gives the existence of the minimizer $X_n$ associated with $\alpha_{p_n}(\mc{L})$. By compactness, we further assume that $X_n$ converges to some $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$. If $X = {\bf 1}$, similarly to the proof of Lemma \ref{lem:extremal_func} above,
we write $X_n = {\bf 1} + Y_n$ and find 
\begin{align*}
  \alpha_1(\mc{L}) - \varepsilon + \frac{1}{k} \ge   \liminf_{n \to \infty} \alpha_{p_n}(\mc{L}) = \liminf_{n \to \infty} - \frac{p_n}{2} \frac{\langle Y_n, \mc{L} Y_n \rangle_{\sigma,\varphi_{p_n}} + O(\norm{Y_n}_{1,\sigma}^3)}{\norm{Y_n}^2_{\sigma,\varphi_{p_n}} + O(\norm{Y_n}_{1,\sigma}^3)} \ge \frac{1}{2}\lambda(\mc{L})\,,
\end{align*}
which again contradicts with Lemma \ref{lem:aux_func}, by letting $k \to \infty$. If $X \neq {\bf 1}$, by definition \eqref{ineq_becp} and \eqref{auxeq:seq}, we have 
\begin{align*}
     \mc{E}_{p_n,\mc{L}}(X_{n}) \le \Big(\alpha_1(\mc{L}) - \varepsilon + \frac{1}{k}\Big) \frac{\norm{X_{n}}_{p_n,\sigma}^{p_n}- 1}{p_n-1}\,,\quad  \text{for}\ n \ \text{large enough}\,.
\end{align*}
It implies that, by letting $n \to \infty$ and $k \to \infty$, and using \eqref{eq:dev_pnorm} with an elementary analysis, 
\begin{align*}
     \mc{E}_{1,\mc{L}}(X) \le \big(\alpha_1(\mc{L}) - \varepsilon\big) \Ent_{1,\sigma}(X)\,.
\end{align*}
It clearly contradicts the optimality of $\alpha_1(\mc{L})$. The proof is complete. 
\end{proof}


\subsubsection{Stability of $\alpha_p(\mc{L})$} We proceed to investigate the stability of the quantum Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state. We will compare the constants $\alpha_p(\mc{L})$
for the following two generators $\mc{L}_\sigma$ and $\mc{L}_{\sigma'}$ that satisfy the detailed balance conditions with respect to two different but commuting full-rank states $\sigma$ and $\sigma'$:
\begin{align} \label{eq:gen_1}
        \mc{L}_\sigma(X) = \sum_{j = 1}^{J} \big(e^{-\omega_j/2}V_j^*[X,V_j] + e^{\omega_j/2}[V_j,X]V^*_j\big)\,,
\end{align}
and 
\begin{align} \label{eq:gen_2}
        \mc{L}_{\sigma'}(X) = \sum_{j = 1}^{J} \big(e^{-\nu_j/2}V_j^*[X,V_j] + e^{\nu_j/2}[V_j,X]V^*_j\big)\,,
\end{align}
where $e^{-\omega_j}$ and $e^{-\nu_j}$ are the eigenvalues of $\Delta_\sigma$ and $\Delta_{\sigma'}$, respectively. For ease of  exposition, we assume that the states $\sigma$ and $\sigma'$ admit the spectral decompositions: 
\begin{align} \label{auxeq_specdecomp}
    \sigma = \sum_{k = 1}^{d} \sigma_k v_k\,,\quad  \sigma' = \sum_{k = 1}^{d} \sigma'_k v_k\,,
\end{align}
respectively. 

\begin{theorem} \label{thm:beck_stab}
Let $\mc{L}_\sigma$ and $\mc{L}_{\sigma'}$ be the generators of two primitive QMS satisfying $\sigma$-DBC and $\sigma'$-DBC, given in \eqref{eq:gen_1} and \eqref{eq:gen_2}, respectively.
Then it holds that 
\begin{align*}
\frac{\Lambda_{\min}}{\Lambda_{\max}} \min_{j} e^{-|\omega_j-\nu_j|(2-p)/2p} \alpha_p(\mc{L}_{\sigma'}) \le \alpha_p(\mc{L}_\sigma)\,,
\end{align*}
where constants $\Lambda_{\min}$ and $\Lambda_{\max}$ are defined as  
\begin{align} \label{eq:const_state}
 \Lambda_{\min} = \min_k \frac{\sigma_k}{\sigma'_k}\,,    \quad \text{and} \quad \Lambda_{\max} = \max_k \frac{\sigma_k}{\sigma'_k}\,.
\end{align}
\end{theorem}

\begin{proof}
Let $X := \Gamma_{\sigma'}^{-1}(\rho) \ge 0$ for $\rho \in \mathcal{D}(\mc{H})$.  We first establish a comparison result for the quantum $p$--divergence $\mc{F}_{p,\sigma}(\rho)$. We define the map
\begin{align*}
    \Phi(A): = \mm
    \Lambda_{\max}^{-1} \Gamma_\si \Gamma_{\sigma'}^{-1} (A) & 0 \\ 0 & 
    \tr(A) - \Lambda_{\max}^{-1}\norm{\Gamma_{\sigma'}^{-1}(A)}_{1,\sigma}
   
    \nn : \ \mathcal{B}(\mc{H}) \to \mc{B}(\mc{H}\oplus \mathbb{C})\,.
\end{align*}
It is easy to check from \eqref{eq:const_state} that $\tr \big(  \Lambda_{\max}^{-1} \Gamma_\si \Gamma_{\sigma'}^{-1} (A) \big) \le \tr (A) $ for any $A \in \mathcal{B}(\mc{H})$. Hence $\Phi$ is completely positive and trace preserving by Kraus representation theorem. Then by the data processing inequality \eqref{eq:dpi_divi}, we have 
\begin{align} \label{eq:dpi}
    \mc{F}_{p, \Phi(\sigma')}\left(\Phi(\rho)\right) \le \mc{F}_{p,\sigma'}\left(\rho\right) = \frac{1}{p(p-1)} \left( \norm{X}_{p,\sigma'}^p - 1 \right). 
\end{align}
We now compute, by definition,
\begin{align} \label{eq:auxdp_2}
      \mc{F}_{p, \Phi(\sigma')}\left(\Phi(\rho)\right) = \frac{1}{p(p-1)} \left( 
   \Lambda^{-1}_{\max} \norm{X}_{p,\sigma}^p + \big(1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}\big)^p \left(1 - \Lambda^{-1}_{\max}\right)^{1-p} - 1
     \right).
\end{align}
Note from \eqref{eq:const_state} that $\Lambda_{\max} \ge 1 \ge \Lambda_{\min}$ and $\Lambda^{-1}_{\max} \norm{X}_{1,\sigma} \le 1$. 
Then  the convexity of $x^p$, $1 < p \le 2$, gives
\begin{align*}
   &(1 - \Lambda^{-1}_{\max})  \left(\frac{1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}}{1 - \Lambda^{-1}_{\max}} \right)^p +   \Lambda^{-1}_{\max} \left(\frac{ \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}}{\Lambda^{-1}_{\max}} \right)^p \\
   = &     \big(1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}\big)^p \big(1 - \Lambda^{-1}_{\max}\big)^{1-p} + \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}^p  \ge 1\,,
\end{align*}
which, by \eqref{eq:auxdp_2}, implies
\begin{align} \label{auxeq_stab}
     \mc{F}_{p, \Phi(\sigma') }\left(\Phi(\rho) \right) \ge \frac{1}{p(p-1)} \left( 
   \Lambda^{-1}_{\max} \norm{X}_{p,\sigma}^p - \Lambda^{-1}_{\max} \norm{X}^p_{1,\sigma}
     \right)\,.
\end{align}
Combining \eqref{eq:dpi} and \eqref{auxeq_stab}, we can find
\begin{equation} \label{eq:sta_main_est_1}
    \frac{1}{p(p-1)} \left( 
    \norm{X}_{p,\sigma}^p -  \norm{X}^p_{1,\sigma}
     \right) \le \frac{\Lambda_{\max}}{p(p-1)} \left( \norm{X}_{p,\sigma'}^p - 1  \right)\,.
\end{equation}




We next give the comparison result for $\mc{E}_{p,\mc{L}}$. We recall \eqref{eq:rep_epl}, and, by Lemma \ref{lem:mono_norm}, obtain
\begin{align} \label{est:p-diri}
     \mc{E}_{p,\mc{L}_\sigma}(X) \ge \left(\inf_{j} e^{-|\omega_j - \nu_j|(2-p)/2p}\right) \frac{p^2}{4} \sum_{j = 1}^J \left\langle \Gamma_\si^{1/p}(\partial_j X), f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p}(X)\right)  \Gamma_\si^{1/p} (\partial_j X)   \right\rangle,
\end{align}
since 
\begin{equation*}
e^{\pm \omega_j/2p} \Gamma_\si^{1/p}(X) \le (\max_j e^{|\omega_j - \nu_j|/2p}) e^{\pm \nu_j/2p} \Gamma_\si^{1/p}(X)\,.    
\end{equation*}
By the integral representation \eqref{eq:integral_thetap} of $f_p^{[1]}$, we can estimate
\small
\begin{align}  \label{auxeqq_stabbdiri}
& \Gamma_\si^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\ = &   \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} 
\Gamma_\si^{1/p} g_0^{[1]}\left(s + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p}
\ ds  \notag \\
 \ge & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \Gamma_\si^{1/p} g_0^{[1]}\left(s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p}\ ds\,, \notag 
\end{align}
\normalsize
where we used the following observation from \eqref{auxeq_specdecomp} and \eqref{eq:const_state}:
\begin{equation} \label{eq:simple_ob}
\Lambda_{\max}^{-1/p} \sigma^{1/p} (\sigma')^{-1/p} \le {\bf 1} \le \Lambda_{\min}^{-1/p} \sigma^{1/p} (\sigma')^{-1/p}\,,     
\end{equation}
and the operator monotonicity of $t^{-1}$. The observation \eqref{eq:simple_ob} also implies that $\Lambda_{\min}^{1/p} \Gamma_\si^{-1/p}\Gamma_{\sigma'}^{1/p}$ is completely positive and trace non-increasing. Then, by \cite[Proposition 3.6]{junge2019stability}, it follows that 
\small
\begin{multline*}
\Gamma_\si^{1/p} g_0^{[1]}\left(s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\
\ge \Lambda_{\min}^{2/p} \Gamma_{\sigma'}^{1/p} g_0^{[1]}\left(s + \Lambda_{\min}^{1/p} e^{\nu_j/2p} \Gamma_{\sigma'}^{1/p}(X), s  + \Lambda_{\min}^{1/p} e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}\,. 
\end{multline*}
\normalsize
Therefore, by \eqref{eq:integral_thetap} and \eqref{auxeqq_stabbdiri}, we have 
\begin{align*}
& \Gamma_\si^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\
\ge & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \Lambda_{\min}^{2/p} \Gamma_{\sigma'}^{1/p} g_0^{[1]}\left(s + \Lambda_{\min}^{1/p} e^{\nu_j/2p} \Gamma_{\sigma'}^{1/p}(X), s  + \Lambda_{\min}^{1/p} e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}
\ ds \\
\ge & \Lambda_{\min} \Gamma_{\sigma'}^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_{\sigma'}^{1/p}(X), e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}\,. 
\end{align*}
Combining the above estimate with \eqref{est:p-diri} and recalling \eqref{eq:rep_epl}, we readily have
\begin{align} \label{eq:sta_main_est_2}
     \mc{E}_{p,\mc{L}_\sigma}(X) \ge \left(\inf_{j} e^{-|\omega_j - \nu_j|(2-p)/2p}\right) \Lambda_{\min} \mc{E}_{p,\mc{L}_{\sigma'}}(X)\,.
\end{align}
The proof is completed by the following simple estimate, with the help of \eqref{eq:sta_main_est_1} and \eqref{eq:sta_main_est_2},
\begin{align*}
     \frac{1}{p-1} \left( 
    \norm{X}_{p,\sigma}^p -  \norm{X}_{1,\sigma}
     \right) & \le \frac{\Lambda_{\max}}{\alpha_p(\mc{L}_{\sigma'})} \mc{E}_{p,\mc{L}_{\sigma'}}(X) \\
     & \le \frac{\Lambda_{\max}}{\alpha_p(\mc{L}_{\sigma'})} \left(\inf_{j} e^{-|\omega_j-\nu_j|(2-p)/2p}\right)^{-1} \Lambda_{\min}^{-1}  \mc{E}_{p,\mc{L}_\sigma}(X)\,. \qedhere
\end{align*}
\end{proof}

\subsection{Applications and examples} \label{sec:app_exp}

This section is devoted to the applications of the $p$-Beckner's inequalities. We first analyze the Beckner constant for the depolarizing semigroup. We then derive an improved bound on the mixing time of quantum Markov dynamics in terms of the Beckner constants $\alpha_p(\mc{L})$. For the symmetric semigroups, by borrowing the techniques from \cite{adamczak2022modified,junge2015noncommutative}, we obtain the moment estimates from Beckner's inequalities \eqref{ineq_becp}, which further allows us to derive concentration inequalities. 



\subsubsection{Beckner constant for depolarizing semigroups} In general, it is challenging to explicitly compute or estimate the optimal constant for the functional inequalities, even in the classical setting. In this section, we consider the quantum Beckner constant $\alpha_p$ for the simplest QMS: the depolarizing semigroup \eqref{def:depolt} with $\gamma = 1$ and $\sigma = {\bf 1}/d$, and show that in this case the computation of $\alpha_p$ is equivalent to the classical one for a random walk. We mention that the explicit values of the log-Sobolev constant $\beta$ and the modified log-Sobolev constant $\alpha_1$ for $\mc{L}_{depol}$ with a general invariant state $\sigma \in \mathcal{D}_+(\mc{H})$ have been obtained in \cite{beigi2020quantum} and \cite{muller2016relative}, respectively. 

\begin{proposition} \label{prop:beck_depol}
Let $\mc{L}_{depol}(X) = \tr(X/d){\bf 1} - X$ be the Lindbladian of the depolarizing semigroup. Then we have 
\begin{align} \label{eq:const_beck}
    \alpha_p(\mc{L}_{depol})  = \inf_{\substack{\theta x + (1 - \theta) y = 1 \\ x,y \ge 0,\  \theta \in \{\frac{1}{d},\ldots, 1 - \frac{1}{d}\} }} \frac{p^2}{4}  \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,.
\end{align}
\end{proposition}

\begin{proof} 
We first compute from definition that 
\begin{equation} \label{auxeq:expbeck}
\alpha_{p}(\mc{L}_{depol}) = \inf_{X \ge 0} \frac{(p-1)\mc{E}_{p,\mc{L}_{depol}}(X)}{\norm{X}_{p,\frac{{\bf 1}}{d}}^p - \norm{X}_{1,\frac{{\bf 1}}{d}}^p} = \frac{p^2}{4} \inf_{X \ge 0} \frac{\norm{X}_{p,\frac{{\bf 1}}{d}}^p- \norm{X}_{p-1,\frac{{\bf 1}}{d}}^{p-1}\norm{X}_{1,\frac{{\bf 1}}{d}} }{\norm{X}_{p,\frac{{\bf 1}}{d}}^p - \norm{X}_{1,\frac{{\bf 1}}{d}}^p}\,.
\end{equation}
We only consider $p \in (1,2)$, since the case $p = 2$, corresponding to the spectral gap, is trivial. Let $\mu_i \ge 0$, $1 \le i \le d$, be the eigenvalues of $X \ge 0$. It is easy to reformulate \eqref{auxeq:expbeck} as
\begin{align*}
\alpha_{p}(\mc{L}_{depol}) = \frac{p^2}{4} \inf_{\mu_i \ge 0}  \frac{ \sum_i \mu_i^p - d^{-1} \big(\sum_{i} \mu_i^{p-1}\big)\big(\sum_i \mu_i\big)}{\sum_i \mu_i^p - d^{-(p-1)}\big(\sum_i \mu_i\big)^p }\,,
\end{align*}
which is equivalent to, for any $\mu_i \ge 0$,  
\begin{equation} \label{auxeq:expbeck_2}
F(\mu_1,\ldots, \mu_d): =   \sum_i \mu_i^p - d^{-1} \big(\sum_{i} \mu_i^{p-1}\big)\big(\sum_i \mu_i\big) - \frac{4 \alpha_{p}(\mc{L}_{depol}) }{p^2} \Big(\sum_i \mu_i^p - d^{-(p-1)}\big(\sum_i \mu_i\big)^p  \Big) \ge 0\,.
\end{equation}
Suppose that $(r_i)_{i =1}^d$ achieves the equality in \eqref{auxeq:expbeck_2}. We claim that all $r_i$ are strictly positive. If not, without loss of generality, we assume $r_1 = 0$, $r_2 > 0$, and
$\sum_{i = 2}^{d} r_i  = 1$. Then, from \eqref{auxeq:expbeck_2}, for small enough $\varepsilon > 0$, we have 
\begin{align*}
F(\varepsilon, r_2 - \varepsilon, r_3, \ldots, r_d) = &\sum_{i = 3} r_i^p - d^{-1} \big(\sum_{i = 3}^d r_i^{p-1}\big) - C_p \Big(\sum_{i = 3}^d r_i^p - d^{-(p-1)}  \Big)  \\
& - d^{-1} \big( (r_2 - \varepsilon) ^{p-1} + \varepsilon^{p-1} \big) - (C_p - 1)  \big( (r_2 - \varepsilon)^p + \varepsilon^p \big) \\
= & - d^{-1} \varepsilon^{p-1} + O(\varepsilon) < 0\,,
\end{align*}
where $C_p := 4 \alpha_p/p^2$. 
This fact contradicts the assumption that $(r_i)_{i}$ saturates the equality, so the claim holds. 
Now, by $r_i > 0$ for all $i$, we have $\nabla F (r_1,\ldots,r_d) = 0$, which, by a direct computation, gives the following equations in the variables $r_i$:
\begin{equation} \label{auxeq:expbeck_3}
\big(\frac{p^2}{4} - \alpha_p \big)r_i^{p-1} - \frac{p(p-1)}{4 d}\norm{X}_1 r_i^{p-2} 
  = \frac{p}{4d}\norm{X}_{p-1}^{p-1} - \alpha_p d^{1-p}\norm{X}_1^{p-1}.
\end{equation}
When $\norm{X}_1$ and $\norm{X}_{p-1}$ are fixed, the above equation clearly
has at most two solutions, denoted by $a$ and $b$, which means that $r_i$ takes the value either $a$ or $b$. Let $n$ be the number of $r_i$ equal to $a$. Then, it follows that
 \begin{align*}
\alpha_{p}(\mc{L}_{depol}) & = \inf_{\substack{n a + (d - n) b = 1 \\ a,b \ge 0,\ n \in \{1,\ldots, d-1\} }}  \frac{p^2}{4}  \frac{ (n a^p + (d - n)b^p) - d^{-1}(n a^{p-1} + (d-n)b^{p-1})}{(n a^p + (d-n)b^p) - d^{-(p-1)}} \\  
& =  \inf_{\substack{\theta x + (1 - \theta) y = 1 \\x,y \ge 0,\ \theta \in \{\frac{1}{d},\ldots, 1-\frac{1}{d}\} }} \frac{p^2}{4}  \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,,
\end{align*}
by setting $\theta = n/d$, $x = d a$, and $y = d b$. 
\end{proof}

To connect the expression \eqref{eq:const_beck} with the classical Beckner constant, we consider a Markov chain on $\{0,1\}$ with the transition matrix:
$$
P = \mm \theta & 1- \theta \\ \theta & 1- \theta \nn\,,
$$
which has the invariant measure $\pi(0) = \theta$, $\pi(1) = 1-\theta$. The Beckner constant for this chain is give by \cite[(4.1)]{bobkov2006modified}
\begin{align} \label{eq:rep_beck}
    \alpha_{p,\theta} & = \inf_{f \ge 0} \frac{p}{2} \frac{- \langle f^{p-1}, (P - I) f \rangle_{\pi}}{\pi(f^p) - \pi(f)^p }  \notag \\
    & =  \inf_{\substack{\theta x + (1 - \theta) y = 1 \\x,y \ge 0 }}  \frac{p}{2}  \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,.
\end{align}
Then we can see $\alpha_{p}(\mc{L}_{depol}) = \inf \big\{ \alpha_{p,\theta}\,;\ \theta \in \{\frac{1}{d},\ldots, 1-\frac{1}{d}\}\big\}$. However, although the representation \eqref{eq:rep_beck} is simple, numerical techniques seem still to be necessary to find the explicit values of $ \alpha_{p,\theta}$ and $\alpha_{p}(\mc{L}_{depol})$. 









\subsubsection{Mixing times}  
In this section, we analyze the mixing time of QMS from quantum $p$-Beckner's inequalities and show that the improved bounds on the mixing time may be obtained. 
For a primitive QMS $\mc{P}_t = e^{t\mc{L}}$, we define the $l_1$ mixing time for $\varepsilon > 0$ by
\begin{align} \label{def:l1mix}
t_1(\varepsilon) = \inf\{t > 0\,;\  \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le \varepsilon \quad \text{for all} \ \rho \in \mathcal{D}(\mc{H}) \}\,.
\end{align}
To bound $t_1(\varepsilon)$, we need the following lemma 
extending \cite[Theorem 4.1]{li2020complete} for the symmetric QMS, which characterizes the convergence of QMS in terms of $\sigma$-weighted $p$-norm. 

\begin{lemma} \label{lem:conver_p_norm}
Let $\mc{P}_t$ be a primitive QMS satisfying $\sigma$-DBC. Then it holds that 
\begin{align*}
      \norm{\mc{P}_t(X) - \tr(\sigma X) {\bf 1}}_{p,\sigma} \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \norm{X}_{p,\sigma}^{1-p/2} \sqrt{\frac{2}{p(p-1)} \left(  \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)}\,, \quad  X \ge 0\,,
\end{align*}
where $\alpha_p (\mc{L}) > 0$ is the quantum Beckner constant associated with $\mc{P}_t$. 
\end{lemma}

\begin{proof}
We define, for $X, Y \in \mc{B}_{sa}(\mc{H})$,  
\begin{equation*}
G_{X,Y}(s) := \norm{X + s Y}_{p,\sigma}^p - \frac{p(p-1)}{2} s^2 \norm{X + s Y}_{p,\sigma}^{p-2} \norm{Y}_{p,\sigma}^2.
\end{equation*}
Similarly to \cite[Theorem 4.1]{li2020complete}, by results in \cite{ricard2016noncommutative}, it is easy to prove that $G_{X,Y}''(0) \ge 0$ for any  self-adjoint $X, Y$, which implies that $G_{X,Y}(s)$ is a convex function on $\mathbb{R}$. We now consider $A := \tr(\sigma X) {\bf 1} $ and $B := X_t - \tr(\sigma X) {\bf 1}$ with $X_t = \mc{P}_t(X)$ for $X \ge 0$. Then, by Lemma \ref{lem:prop:norm_power}, we have, for any $s \in \mathbb{R}$,
\begin{equation*}
    \norm{A + s B}_{p,\sigma}^p \ge  \norm{A + s B}_{1,\sigma}^p \ge \tr(\sigma(A + s B))^p  = \tr(\sigma X)^p = \norm{A}_{p,\sigma}^p\,.
\end{equation*}
It then follows from definition that $G_{A,B}'(0) \ge 0$, which, along with the convexity of $G_{A,B}(s)$, yields $G'_{A,B}(s) \ge 0$ for any $s \ge 0$. Hence, we have $G_{A,B}(1) \ge G_{A,B}(0) $, that is (recalling the definitions of $A, B$), 
\begin{equation} \label{auxeq:decay}
     \norm{X_t}_{p,\sigma}^p - \frac{p(p-1)}{2} \norm{X_t}_{p,\sigma}^{p-2} \norm{X_t - \tr(\sigma X) {\bf 1}}_{p,\sigma}^2 \ge \norm{X}_{1,\sigma}^p\,.
\end{equation}
By \eqref{ineq_becp} and Gr\"{o}nwall’s inequality, it holds that 
\begin{align*}
    \norm{X_t}_{p,\sigma}^p - \norm{X_t}_{1,\sigma}^{p} \le e^{- \frac{4 \alpha_p}{p}t} \left(  \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)\,,\quad X \ge 0\,,
\end{align*}
which, along with \eqref{auxeq:decay}, implies 
\begin{align*}
     \norm{X_t - \tr(\sigma X) {\bf 1}}_{p,\sigma}^2 \le \frac{2}{p(p-1)} e^{- \frac{4 \alpha_p}{p}t} \norm{X}_{p,\sigma}^{2-p}\left(  \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)\,.
\end{align*}
The proof is complete by taking the square root of the above inequality. 
\end{proof}

\begin{proposition} \label{prop:mixing}
Under the same assumption as in Lemma \ref{lem:conver_p_norm}, it holds that
\begin{align} \label{eq:mixing}
t_1(\varepsilon) \le \inf_{p \in (1,2]} h(p,\sigma_{\min},\varepsilon)\,,
\end{align}
where 
\begin{align*}
    h(p,\sigma_{\min},\varepsilon) := \frac{p}{2\alpha_p(\mc{L})} \log \bigg( \varepsilon^{-1} \sqrt{\frac{2}{p(p-1)} \left(  \sigma_{\min}^{\frac{2}{p}-2} -  \sigma_{\min}^{ p + \frac{2}{p} -3}  \right)} \bigg)\,.
\end{align*}
\end{proposition}


\begin{proof}
Let $X_t$ be the relative density of $\rho_t = e^{t \mc{L}^\dag}(\rho)$; see \eqref{eq:evo_density}. We write
\begin{align*}
    \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1  = \big\lVert \Gamma_\si (\Gamma_\si^{-1} \mc{P}^\dag_t(\rho) - {\bf 1})\big\rVert_1 = \norm{X_t - {\bf 1}}_{1,\sigma}\,.
\end{align*}
By Lemma \ref{lem:conver_p_norm}, we have 
\begin{align}  \label{eq:sup_pnorm}
  \sup_{\rho \in \mathcal{D}(\mc{H})}  \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \sup_{X \ge 0,\, \tr(\sigma X) = 1} \sqrt{\frac{2}{p(p-1)} \left(  \norm{X}_{p,\sigma}^2 - \norm{X}_{p,\sigma}^{2-p} \right)}\,. 
\end{align}
Recalling the formula \eqref{eq:suprhodp}, by definition of $D_p(\rho \| \sigma) $, there holds 
\begin{align*}
    \sup_{X \ge 1,\, \tr(\sigma X) = 1} \norm{X}_{p,\sigma} = \sigma_{\min}^{\frac{1}{p}-1}\,.
\end{align*}
Also note that the function $x^2 - x^{2 - p}$ with $p \in (1,2]$ is increasing for $x \ge 1$. Then it follows from \eqref{eq:sup_pnorm} that 
\begin{align*}
      \sup_{\rho \in \mathcal{D}(\mc{H})}  \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \sqrt{\frac{2}{p(p-1)} \left(  \sigma_{\min}^{\frac{2}{p}-2} -  \sigma_{\min}^{ p + \frac{2}{p} -3} \right)}\,. 
\end{align*}
By definition \eqref{def:l1mix} and a direct computation, we obtain the estimate \eqref{eq:mixing} for $t_1(\varepsilon)$. 
\end{proof}
By elementary calculus, we find 
\begin{align*}
     h(2,\sigma_{\min},\varepsilon) = \frac{1}{\lambda(\mc{L})} \log \bigg(\varepsilon^{-1} \sqrt{  \sigma_{\min}^{-1} -  1  } \bigg)\,,
\end{align*}
and, as $p \to 1^+$,
\begin{align*}
     h(p, \sigma_{\min}, \varepsilon) \to \frac{1}{2\alpha_1(\mc{L})} \log \bigg(\varepsilon^{-1} \sqrt{  2 \log \big(\sigma_{\min}^{-1}\big)} \bigg)\,,
\end{align*}
which are nothing  but the mixing time bounds obtained from the decays of the variance and the relative entropy, respectively \cite{kastoryano2013quantum,temme2010chi}. It means that $\inf_{p \in (1,2]} h(p,\sigma_{\min},\varepsilon)$ in \eqref{eq:mixing} can be indeed an improved bound for the mixing time in some cases. 

\subsubsection{Moment estimates and concentration inequalities} In this section, we consider the primitive symmetric QMS $\mc{P}_t = \mc{P}_t^\dag$; see Remark \ref{eq:tracial}. We will derive the moment estimates from quantum $p$-Beckner's inequalities, by extending the arguments of \cite[Proposition 3.3]{adamczak2022modified} for classical Markov semigroups to the quantum setting. This helps us to obtain concentration inequalities in a similar manner as \cite{junge2015noncommutative}. We first recall  the carr\'{e} du champ operator (gradient form) associated with $\mc{P}_t = e^{t \mc{L}}$ \cite{junge2015noncommutative,wirth2021curvature}:  
\begin{align} \label{def:gamma}
    \Gamma(X,Y) = \frac{1}{2}(\mc{L}(X^*Y) - X^* (\mc{L}Y) - (\mc{L}X)^*Y)\quad \text{for}\  X,Y \in \mc{B}(\mc{H})\,.
\end{align}
As usual, we write $\Gamma(X)$ for $\Gamma(X,X)$. By the self-adjointness $\mc{L}^\dag = \mc{L}$ and $\mc{L}({\bf 1}) = 0$, a direct computation gives the relation between the $\Gamma$ operator and the Dirichlet form $\mc{E}_{2,\mc{L}}(X,Y) := -\langle X, \mc{L} Y \rangle_{\frac{{\bf 1}}{d}} $: 
\begin{align} \label{eq:rela_gamma}
\mc{E}_{2,\mc{L}}(X,Y) = \big\langle \frac{{\bf 1}}{d}, \Gamma(X,Y)\big\rangle\,.   
\end{align}
Before we state our main result on moment estimates, we need the following useful lemma. 

\begin{lemma}\label{lem:diri_convex}
For any differentiable convex and increasing function $\varphi: [0,\infty) \to \mathbb{R}$ and $c \in \mathbb{R}$, it holds that
\begin{align} \label{eq:diri_convex}
    \mc{E}_{2,\mc{L}}(\varphi(|X + c|), |X + c|) \le 2 \langle\varphi'(|X + c|), \Gamma(X) \rangle_{\frac{{\bf 1}}{d}} \le 2 \norm{\varphi'(|X+c|)}_{p,\frac{{\bf 1}}{d}} \norm{\Gamma(X)}_{\h{p},\frac{{\bf 1}}{d}}\,,
\end{align}
for $X \in \mc{B}_{sa}(\mc{H})$, where $p \ge 1$. 
\end{lemma}
\begin{proof}
Assume that $X \in \mc{B}_{sa}(\mc{H})$ has the spectral decomposition $X = \sum \lambda_i E_i$, where $E_i$ are the eigen-projections associated with the eigenvalues $\lambda_i$. Note from the convexity of $\varphi$ that for any $x,y \ge 0$, 
\begin{align} \label{auxeq:convex_fun}
      \frac{\varphi(x) - \varphi(y)}{x - y} \le \max \{\varphi'(x), \varphi'(y)\} \le \varphi'(x) + \varphi'(y)\,,
\end{align} 
since $\varphi'(x) \ge 0$ holds by the monotonicity of $\varphi$. Recalling the formula \eqref{eq:inte_by_parts}, by the eigen-decomposition of $X$ and the inequality \eqref{auxeq:convex_fun}, we have
\begin{align} \label{auxeq_momen_1}
      \mc{E}_{2,\mc{L}}(\varphi(|X+c|), |X + c|) & = \sum_{j \in J} \big\langle \partial_j \varphi(|X+c|), \partial_j |X + c| \big\rangle_{\frac{{\bf 1}}{d}} \notag \\
      & = \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi(|\lambda_i+c|) - \varphi(|\lambda_k + c|))(|\lambda_i + c| - |\lambda_k+c|) \tr (E_k V_j E_i V_j) \notag \\
      & \le \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi'(|\lambda_i+c|) + \varphi'(|\lambda_k + c|))(|\lambda_i + c| - |\lambda_k+c|)^2 \tr (E_k V_j E_i V_j)\,.
\end{align}
Similarly, by definition \eqref{def:gamma} of $\Gamma(X)$, we also compute 
\begin{align} \label{auxeq_momen_2}
  2 \big\langle\varphi'(|X+c|), \Gamma(X) \big\rangle_{\frac{{\bf 1}}{d}} = & \frac{1}{d} \sum_{j \in J} \sum_{i,k} \big\{(\varphi'(|\lambda_i+c|) - \varphi'(|\lambda_k + c|))(\lambda^2_i - \lambda_k^2) \notag \\
  & \qquad \qquad -  2(\lambda_i\varphi'(|\lambda_i + c|) - \lambda_k \varphi'(|\lambda_k + c|))(\lambda_i - \lambda_k) \big\} \tr (E_k V_j E_i V_j) \notag \\
  = & \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi'(|\lambda_i+c|) + \varphi'(|\lambda_k + c|))(\lambda_i - \lambda_k)^2 \tr (E_k V_j E_i V_j)\,.
\end{align}
Since there holds $\big||\lambda_i + c| - |\lambda_k+c| \big| \le |\lambda_i - \lambda_k|$, by \eqref{auxeq_momen_1} and \eqref{auxeq_momen_2}, and using H\"{o}lder's inequality, we obtained the desired estimate \eqref{eq:diri_convex}. 
\end{proof}

\begin{proposition} \label{prop:moment}
Suppose that the quantum $p$-Beckner's inequality \eqref{ineq_becp} holds
for all $p \in (1,2]$ with $\alpha_p \ge a(p-1)^s$ for some $a > 0$ and $s \ge 0$. Then we have, for $X \in \mc{B}_{sa}(\mc{H})$ and $r \ge 2$,
\begin{align} \label{eq:moment}
    \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le \frac{r^{s + 1}\kappa(s)}{a}\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,,
\end{align}
where $\kappa(s) := (1 - e^{-(s+1)/2})^{-1}$. 
\end{proposition}
\begin{proof}
We shall prove by induction that for all positive integers $k$ and $r \in (k,k+1]$, there holds 
\begin{align} \label{eq:ineq_induc}
     \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le c_r \norm{\Gamma(X)}_{\max\{\frac{r}{2},1\},\frac{{\bf 1}}{d}}\,, \quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align}
where 
\begin{align} \label{def:constant_moment}
    c_r := \frac{r^{s + 1}\kappa_r(s)}{a} \,, \quad \kappa_r(s) := \Big( 1 - \big(  \frac{r-1}{r} \big)^{(s+1)r/2}\Big)^{-1}\,.
\end{align}
The desired estimate \eqref{eq:moment} is a direct consequence of \eqref{eq:ineq_induc}, since $\kappa_r$ increases in $r$ and $\kappa_r(s) \to \kappa(s)$ as $r \to \infty$.  We first note that by Theorem \ref{thm:beck_poincare}, \eqref{ineq_pi} holds with $\lambda(\mc{L}) \ge a$. For $k = 1$ and $r \in (1,2]$, by  \eqref{ineq_pi} and \eqref{eq:rela_gamma}, we have
\begin{align*}
 \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{2,\frac{{\bf 1}}{d}}^2 \le \frac{1}{\lambda(\mc{L})} \mc{E}_{2,\mc{L}}(X) \le c_r \norm{\Gamma(X)}_{1,\frac{{\bf 1}}{d}}\,,
\end{align*}
since $c_r$ is increasing in $r$, which gives $c_r \ge c_1 = 1/a \ge 1/\lambda(\mc{L})$. Suppose that \eqref{eq:ineq_induc} holds for all integers smaller than some $k > 1$. Now we consider $r \in (k,k+1]$. We define $Y = | X - \norm{X}_{1,\frac{{\bf 1}}{d}}|$, and then estimate by using \eqref{ineq_becp},
\begin{align} \label{auxeqpr_1}
\alpha_{\h{r}}\big(\norm{Y^{r-1}}_{\h{r},\frac{{\bf 1}}{d}}^{\h{r}} -  \norm{Y^{r-1}}_{1,\frac{{\bf 1}}{d}}^{\h{r}}\big) \le (\h{r} - 1)\mc{E}_{\h{r},\mc{L}}(Y^{r-1})\,.
\end{align}
By Lemma \ref{lem:diri_convex} with $\varphi(x) = x^{r - 1}$, $c = - \norm{X}_{1,\frac{{\bf 1}}{d}}$, and $p = r/(r-2)$, we have 
\begin{align} \label{auxeqpr_2}
    (\h{r} - 1)\mc{E}_{\h{r},\mc{L}}(Y) = - \frac{\h{r}^2}{4}\langle Y^{r-1}, \mc{L}Y \rangle_{\frac{{\bf 1}}{d}} & \le \frac{\h{r}^2}{2}(r - 1)\norm{Y^{r-2}}_{p,\frac{{\bf 1}}{d}} \norm{\Gamma(X)}_{\h{p},\frac{{\bf 1}}{d}} \notag \\
    & = \frac{1}{2} \frac{r^2}{r-1} \norm{Y}_{r,\frac{{\bf 1}}{d}}^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,.
\end{align}
We write $l_r = \norm{Y}_{r,\frac{{\bf 1}}{d}}$ and note $\alpha_{\h{r}} \ge a(\h{r} - 1)^s = a (r - 1)^{-s}$. Then combining estimates \eqref{auxeqpr_1} and \eqref{auxeqpr_2} gives
\begin{align} \label{auxeqpr_3}
     l_r^r - l_{r-1}^{r} \le \frac{r^2 (r-1)^{s-1}}{2a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}} \le \frac{r^{s+1}}{a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,, 
\end{align}
since $ r/(r-1)\le 2$. Applying the assumption \eqref{eq:ineq_induc} to bound $l_{r-1}$ and by \eqref{auxeqpr_3}, we obtain 
\begin{align} \label{auxeqpr_4}
     l_r^r \le \big(c_{r-1}\norm{\Gamma(X)}_{\max\{\frac{r-1}{2},1\},\frac{{\bf 1}}{d}}\big)^{\frac{r}{2}} + \frac{r^{s+1} }{a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,.
\end{align}
Note from \eqref{def:constant_moment} that 
\begin{align} \label{auxeqpr_5}
    \frac{c_{r-1} \norm{\Gamma(X)}_{\max\{\frac{r-1}{2},1\},\frac{{\bf 1}}{d}}}{c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}} \le \frac{c_{r-1}}{c_r} \le \frac{(r-1)^{s+1}\kappa_{r-1}(s)}{r^{s+1}\kappa_r(s)}\,.
\end{align}
By dividing both sides of \eqref{auxeqpr_4} by $(c_r\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}})^{r/2}$ and using \eqref{auxeqpr_5}, it follows that 
\begin{align} \label{auxeqpr_6}
    \Big( \frac{l_r^2}{c_r\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}}\Big)^{\frac{r}{2}} \le \Big( \frac{r-1}{r} \Big)^{\frac{(s+1)r}{2}} + \frac{1}{\kappa_r} \Big( \frac{l_r^2}{c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}} \Big)^{\frac{r-2}{2}}\,.
\end{align}
To complete the proof, we need the fact from \cite[Proposition 3.1]{adamczak2022modified} that the function $h(x) = (1-1/r)^{(1+s)r/2} + \kappa_r^{-1} x^{1-2/r} - x$ is strictly concave on $[0,\infty)$ and satisfies $h(0) > 0$ and $h(1) = 0$, which means that $h(x) \ge 0$ implies $x \le 1$. This fact, along with \eqref{auxeqpr_6}, readily implies 
\begin{equation*}
   \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,. \qedhere
\end{equation*}
\end{proof}
\begin{remark}
In the case of $s = 0$, Proposition \ref{prop:moment} shows that if \eqref{ineq_becp} holds with $\inf_{p\in(1,2]}\alpha_p \ge a$, then we have
\begin{align*} 
    \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}^2_{r,\frac{{\bf 1}}{d}} \le  \frac{r\kappa}{a } \norm{\Gamma(X)}_{\infty}\,,\quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align*} 
where $\kappa := (1 - e^{-1/2})^{-1}$, since $\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}} \le  \norm{\Gamma(X)}_{\infty}$.  A similar result was obtained by Junge and Zeng \cite{junge2015noncommutative} for non-primitive symmetric QMS under the Bakry-\'{E}mery curvature-dimension condition ($\Gamma_2$-criterion): 
\begin{align}\label{def:becdc}
 \Gamma_2(X,X) \ge \alpha \Gamma(X,X)\,, \quad \forall X \in \mathcal{B}(\mc{H})\,, \tag{BE$(\alpha,\infty)$}
\end{align}
for some $\alpha > 0$, where $\Gamma_2$ is the iterated carr\'{e} du champ operator: $\Gamma_2(X,Y): = - \frac{1}{2}(\Gamma(X,\mc{L} Y) + \Gamma(\mc{L}X,Y) - \mc{L}\Gamma(X,Y))$. To be precise, they used the noncommutative martingale methods and proved that
under some necessary regularity condition, if \ref{def:becdc} holds, then we have
\begin{align*}
    \norm{X - E(X)}_r^2 \le \frac{8r}{\alpha} \norm{\Gamma(X)}_{\infty}\,,\quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align*}
where $E$ is the conditional expectation to the fixed point algebra $\{X\,;\ \mc{P}_t(X)= X\ \text{for}\  t \ge 0\}$. It is currently unknown whether or not the quantum Beckner's inequality can be implied from $\Gamma_2$-criterion \ref{def:becdc}.  Hence, our result is complementary with theirs.  
\end{remark}
 
Similarly to \cite[Corollary 4.13]{junge2015noncommutative}, we next show that the moment estimates \eqref{eq:moment} with $s= 0$ can imply a noncommutative exponential integrability and an associated Gaussian concentration inequality. The idea of the proof is borrowed from the commutative case \cite{ben2008poincare} by Efraim and Lust-Piquard. 


\begin{corollary} \label{cor:concern}
Suppose that the quantum $p$-Beckner's inequality \eqref{ineq_becp} holds
 with $\inf_{p\in(1.2]}\alpha_p \ge a$ for some $a > 0$.
 Then it holds that  
\begin{equation} \label{eq:ex_int}
    \norm{\exp\big(\big|X- \norm{X}_{1,\frac{{\bf 1}}{d}}\big|\big)}_{1,\frac{{\bf 1}}{d}} \le 2 \exp\bigg(  \frac{e\kappa \norm{\Gamma(X)}_{\infty}}{2 a}  \bigg)\,,
\end{equation}
and, for any $t > 0$,
\begin{align} \label{eq:concentration}
      \frac{1}{d}\tr\Big(1_{[t,\infty)}\big(\big|X- \norm{X}_{1,\frac{{\bf 1}}{d}}\big|\big)\Big) \le 2 \exp \left( -
      \frac{a t^2}{2 e \kappa  \norm{\Gamma(X)}_{\infty}}
      \right)\,,
\end{align}
where $\kappa = (1 - e^{-1/2})^{-1}$. 
\end{corollary}



\begin{proof}
Note that $\Gamma(X) = \Gamma\big(X - \norm{X}_{1,\frac{{\bf 1}}{d}}\big)$. Without loss of generality, we assume $\norm{X}_{1,\frac{{\bf 1}}{d}} = 0$.  By functional calculus and
Proposition \ref{prop:moment}, we obtain 
\begin{align}  \label{auxeq:ex_int}
    \frac{1}{2d} \tr (e^{|X|}) &\le  \frac{1}{d}\tr(\cosh X) = 1 + \sum_{j = 1}^\infty \frac{1}{(2j)!}\norm{X}^{2j}_{2j, \frac{{\bf 1}}{d}} \notag \\
    & \le 1 + \sum_{j = 1}^\infty \frac{1}{(2j)!} \frac{(2j)^{j}\kappa^j}{a^j}\norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}}^j \notag \\
    & \le 1 + \sum_{j = 1}^\infty \frac{j^j}{j! (2 j -1)!!} \bigg( \frac{\kappa \norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}} }{a} \bigg)^j \notag \\
    & \le 1 + \sum_{j = 1}^\infty \frac{1}{j!} \bigg( \frac{e\kappa \norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}}}{2a} \bigg)^j\,, 
\end{align}
where in the last inequality we have used $\frac{j^j}{(2j-1)!!} \le \left( \frac{e}{2} \right)^j$ for all $j \in \mathbb{N}$ from Stirling's formula. Then the inequality \eqref{eq:ex_int} follows from \eqref{auxeq:ex_int} and again $\norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}} \le \norm{\Gamma(X)}_\infty$. For the concentration inequality \eqref{eq:concentration}, by \eqref{eq:ex_int} and Chebyshev inequality, we have, for any $\lambda > 0$, 
\begin{align*}
    \frac{1}{d}\tr\big(1_{[t,\infty)}(|X|)\big) \le e^{-\lambda t} \frac{1}{d} \tr \big( e^{\lambda |X|} \big) \le 2 \exp\bigg(  \frac{e\kappa \norm{\Gamma(X)}_{\infty}}{2a} \lambda^2 - t \lambda \bigg)\,.
\end{align*}
Then, choosing $\lambda$ to minimize the right-hand side of the above estimate gives the desired \eqref{eq:concentration}.
\end{proof}

\begin{remark}
Bobkov and G\"{o}tze \cite{bobkov1999exponential} established the exponential integrability in the commutative case from a variant of log-Sobolev inequality. Moreover, they showed that the exponential integrability implies the transportation cost inequality of order $1$. A noncommutative analog of this result can be found in \cite[Corollary 4.19]{junge2015noncommutative}. Also note that Rouz\'{e} and Datta \cite{rouze2019concentration} showed a similar Gaussian concentration inequality from the transportation cost inequalities that can be implied by MLSI.  Corollary \ref{cor:concern} above provides another approach from the quantum MLSI to the concentration inequality, in view of Corollary \ref{coro:pitobeck}.
\end{remark}















\section{Generalized quantum optimal transport} \label{sec:qot_beck}


\subsection{Gradient flow of quantum \texorpdfstring{$p$} --divergence} \label{sec:gradient}

This section aims to identify the dual QMS $\mc{P}_t^\dag = e^{t \mc{L}^\dag}$ with $\sigma$-DBC as the gradient flow of the quantum $p$--divergence $\mc{F}_{p,\sigma}$, for any $p \in (1,2]$, with respect to the Riemannian metrics $g_{p,\mc{L}}$ constructed in \eqref{def:metric_tensor} below. The arguments follow closely with those in \cite{carlen2017gradient} (see also \cite{otto2001geometry} for the classical case). 

For our purpose, we first compute the functional derivative $\delta_\rho \mc{F}_{p,\sigma}(\rho)$ of $\mc{F}_{p,\sigma}$ by 
\begin{align*}
    \frac{d}{d t}\Big|_{t = 0} \mc{F}_{p,\sigma}(\rho_t)  = \langle  \delta_\rho \mc{F}_{p,\sigma}(\rho), \dot{\rho}\rangle\,,
\end{align*}
for any smooth curve $\rho_t: (-\varepsilon,\varepsilon) \to \mc{D}(\mc{H})$, $\varepsilon > 0$, satisfying $\rho_0 = \rho$. Similarly to the computation for \eqref{eq:ep_divi}, we find 
\begin{equation} \label{eq:deriv_func}
    \delta_\rho \mc{F}_{p,\sigma}(\rho) = \frac{1}{p-1} \Gamma_\si^{-1/\h{p}}\Big(\left(\Gamma_\si^{-1/\h{p}}(\rho)\right)^{p-1}\Big)\,, \quad \forall \rho \in \mc{D}(\mc{H})\,.
\end{equation}
Then, letting $Y := \Gamma_\si^{-1/\h{p}}(\rho)$, by \eqref{eq:partialpower_2} and Lemma \ref{lem:chain_rule},
we derive, for each $1 \le j \le J$, 
\begin{align} \label{eq:deriv_entropy}
    \partial_j  \delta_\rho \mc{F}_{p,\sigma}(\rho) & = \frac{1}{p-1} \partial_j\Gamma_\si^{-1/\h{p}}\left(Y^{p-1}\right) \notag \\ & = \frac{1}{p-1}\Gamma_\si^{-1/\h{p}}\left(V_j \left(e^{-\omega_j/2p} Y\right)^{p-1} - \left(e^{\omega_j/2p} Y\right)^{p-1} V_j\right) \notag \\
    & = \Gamma_\si^{-1/\h{p}} \left(f_p^{[1]}\left(e^{\omega_j/2p} Y,  e^{-\omega_j/2p} Y \right) \left(V_j \left(e^{-\omega_j/2p} Y \right) - \left(e^{\omega_j/2p} Y\right) V_j \right)\right),
\end{align}
where $f_p$ is given in \eqref{def:funfp}.
Again by \eqref{eq:partialpower_2} and noting $\partial_{j,\sigma} = \Gamma_\si \partial_j \Gamma_\si^{-1}$ from \eqref{eq:adjpjkms}, it follows that
\begin{align} \label{eq:deriv_entropy_aux}
    V_j \left(e^{-\omega_j/2p} Y\right) - \left(e^{\omega_j/2p} Y\right) V_j &= \Gamma_\si^{1/p} \left( \partial_j \Gamma_\si^{-1/p} (Y) \right)  \notag \\ & =  \Gamma_\si^{1/p} \left( \partial_j \Gamma_\si^{-1} (\rho) \right) =  \Gamma_\si^{-1/\h{p}} (\partial_{j,\sigma}\rho)\,.
\end{align}
Combining \eqref{eq:deriv_entropy} and \eqref{eq:deriv_entropy_aux} readily gives
\begin{align} \label{eq:double_deriv}
      \partial_j  \delta_\rho \mc{F}_{p,\sigma}(\rho) = \Gamma_\si^{-1/\h{p}} \left(f_p^{[1]}\left(e^{\omega_j/2p} \Gamma_\si^{-1/\h{p}}(\rho),  e^{-\omega_j/2p}  \Gamma_\si^{-1/\h{p}}(\rho)\right)\left(\Gamma_\si^{-1/\h{p}} (\partial_{j,\sigma} \rho)\right)\right).
\end{align}

We next define the Riemannian structure, associated with $\mc{F}_{p,\sigma}(\rho)$, on the matrix manifold $\mc{D}_+(\mc{H})$, i.e., a family of inner products on the tangent space $T_\rho = \mc{B}^0_{sa}(\mc{H}) := \{X \in \mc{B}_{sa}(\mc{H})\,;\ \tr(X) = 0\}$ that depends smoothly on $\rho$. 
Motivated by the formula \eqref{eq:double_deriv}, we first define the operator $ [\rho]_{p,\omega}: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})$ for $\rho \in \mc{D}_+(\mc{H})$ and $\omega \in \mathbb{R}$ by
\begin{align} \label{def:kernel_multiplication}
    [\rho]_{p,\omega} = \Gamma_\si^{1/\h{p}} \circ
    \theta_p\left(e^{\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho),  e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho)\right) \circ
    \Gamma_\si^{1/\h{p}}\,,
\end{align}
where
\begin{align} \label{def:theta_p}
     \theta_p(x,y) := \big(f_p^{[1]}(x,y)\big)^{-1}
     = \left\{
    \begin{aligned}
  &  (p-1) \frac{x - y}{x^{p-1} - y^{p-1}}\,, \quad & x \neq y\,, \\
   &  x^{2-p} \,,\quad  & x = y\,.
    \end{aligned}
    \right.
\end{align}
It is clear that when $\sigma = {\bf 1}/d$ and $\omega = 0$, we have $[\rho]_{p,\omega} A = d^{1-p} \rho^{2-p} A = \Gamma_\si( (\Gamma_\si^{-1}(\rho))^{2-p}) A$ for any $A$ that commutes with $\rho$. Therefore, $ [\rho]_{p,\omega}$ can be regarded as 
a noncommutative analog of the multiplication by the relative density $(\Gamma_\si^{-1}\rho)^{2-p}$ with respect to the reference state $\sigma$. Noting $\Gamma_\si^{-1/\h{p}}(\rho) > 0$, by Lemma \ref{lem:double_inner} the operator $[\rho]_{p,\omega}$ is
evidently invertible. Then, we immediately obtain from \eqref{eq:double_deriv} that
\begin{align} \label{auxeq_deriva}
     \partial_j  \delta_\rho \mc{F}_{p,\sigma}(\rho) = [\rho]_{p,\omega_j}^{-1}\partial_{j,\sigma}\rho\,.
\end{align}
With the help of $[\rho]_{p,\omega}$, we now introduce the operator:
\begin{align} \label{def:dprho} 
\mf{D}_{p,\rho}(A) := \sum_{j = 1}^J \partial_j^\dag \left([\rho]_{p,\omega_j} \partial_j A \right):\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align}
which, as we shall see soon, is crucial for defining the desired Riemannian metrics on $\mc{D}_+(\mc{H})$. The following result summarizes the main properties of $[\rho]_{p,\omega}$ and $\mf{D}_{p,\rho}$ and can be proved in the same manner as \cite[Lemma 5.8]{carlen2017gradient} and \cite[Lemma 7.3]{carlen2020non}. Hence we omit its proof.


\begin{lemma} \label{lem:basic_krho} 
Let $\rho \in \mc{D}_+(\mc{H})$ and $\omega \in \mathbb{R}$. It holds that 
\begin{enumerate}[1.]
    \item  $\langle \cdot, [\rho]_{p,\omega}(\cdot)\rangle$ gives an inner product on $\mathcal{B}(\mc{H})$. Moreover, both $[\rho]_{p,\omega}$ and $[\rho]_{p,\omega}^{-1}$ are $C^\infty$ on  $\mc{D}_+(\mc{H})$. 
    \item  $\mf{D}_{p,\rho}$ is a positive semidefinite operator on $\mathcal{B}(\mc{H})$ and preserves self-adjointness. Moreover, we have 
\begin{align*}
    \ran(\mf{D}_{p,\rho}) = \ran(\mc{L}^\dag) = \ran({\rm div})\,,\quad \ker(\mf{D}_{p,\rho}) = \ker(\mc{L})=\ker(\nabla)\,.
\end{align*} 
\end{enumerate}
\end{lemma}
Recalling $\ker(\mc{L}) = {\rm span}\{{\bf 1}_{\mc{H}}\}$ for a primitive QMS $\mc{P}_t = e^{t \mc{L}}$, a direct consequence of the above lemma is that $\mf{D}_{p,\rho}$ for $\rho \in \mc{D}_+(\mc{H})$ is a positive definite operator on $\mc{B}_{sa}^0(\mc{H}) = (\ker(\mf{D}_{p,\rho}))^\perp$ that depends $C^\infty$ on $\rho$.
These facts allow us to define the following class of Riemannian metrics on $\mc{D}_+(\mc{H})$.
\begin{definition}\label{def:metric}
For each $\rho \in \mc{D}_{+}(\mc{H})$, we define the metric tensor $g_{p,\rho}$ on the tangent space $T_\rho = \mc{B}_{sa}^0(\mc{H})$ by 
\begin{align} \label{def:metric_tensor}
    g_{p,\rho}(\nu_1, \nu_2) := \langle \mf{D}_{p,\rho}^{-1} (\nu_1), \nu_2 \rangle \,,\quad \nu_1, \nu_2 \in T_\rho\,.
\end{align}
\end{definition}
We introduce $U_i := \mf{D}_{p,\rho}^{-1} (\nu_i) \in \mc{B}_{sa}^0(\mc{H})$ and define the inner product on $\mc{B}(\mc{H})^J$: 
\begin{align*}
    \langle {\bf A}, {\bf B}\rangle_{p,\rho} := \sum_{j = 1}^J \langle A_j, [\rho]_{p,\omega_j} B_j\rangle \quad \text{for} \ {\bf A}, {\bf B} \in \mc{B}(\mc{H})^J\,.
\end{align*}
Then, by \eqref{def:dprho}, the metric tensor $g_{p,\rho}$ can be rewritten as 
\begin{align} \label{def:gp2}
     g_{p,\rho}(\nu_1, \nu_2) = \sum_{j = 1}^{J} \langle \partial_j U_1, [\rho]_{p,\omega_j} \partial_j U_2\rangle = \langle \nabla U_1, \nabla U_2\rangle_{p,\rho}\,.
\end{align}
We are now ready to conclude Proposition \ref{prop:gradient_flow} below. We first recall that the Riemannian gradient of $\mc{F}_{p,\sigma}$ 
with respect to the metric $g_{p,\rho}$, denoted by
$\grad \mc{F}_{p,\sigma}(\rho)$, 
is determined by the relation:
\begin{align*}
 g_{p,\rho}({\rm grad}\mc{F}_{p,\sigma}(\rho), \nu) = \langle \delta_\rho \mc{F}_{p,\sigma}(\rho), \nu\rangle \,, \quad \forall \nu \in T_\rho\,.
\end{align*}
Then, by Definition \ref{def:metric} and formulas \eqref{auxeq_deriva} and \eqref{def:dprho}, it follows that
\begin{align} \label{eq:riemannian_grad}
    {\rm grad} \mc{F}_{p,\sigma}(\rho) = \mf{D}_{p,\rho} (\delta_\rho \mc{F}_{p,\rho}(\rho)) = - \mc{L}^\dag(\rho)\,,
\end{align} 
and the following result holds. 
\begin{proposition}\label{prop:gradient_flow}
The dual primitive QMS, $\rho_t = e^{t \mc{L}^\dag}(\rho)$ for $\rho \in \mathcal{D}_+(\mc{H})$, satisfying $\sigma$-DBC is the gradient flow of the quantum $p$--divergence $\mc{F}_{p,\sigma}$ with respect to the Riemannian metric $g_{p,\rho}$ in \eqref{def:metric_tensor}:
\begin{align*}
    \partial_t{\rho}_t = - {\rm grad} \mc{F}_{p,\sigma}(\rho_t) = \mc{L}^\dag(\rho_t)\,.
\end{align*}
\end{proposition}

An immediate consequence of Proposition  \ref{prop:gradient_flow} above is the decrease of the entropy functional $\mc{F}_{p,\sigma}(\rho)$ along the quantum dynamic $\rho_t = \mc{P}_t^\dag(\rho)$:
\begin{align} \label{eq:grad_entropy}
    \frac{d}{d t} \mc{F}_{p,\sigma}(\rho_t) & = g_{p,\rho_t}({\rm grad} \mc{F}_{p,\sigma}(\rho_t),\dot{\rho}_t) = - g_{p,\rho_t}(\mc{L}^\dag(\rho_t), \mc{L}^\dag(\rho_t)) \le 0\,.
\end{align} 
We shall see later in Section \ref{sec:geodesic_convexity} that the decay rate of $\mc{F}_{p,\sigma}(\rho_t)$, characterized by the quantum Beckner's inequality (cf.\,\eqref{eq:exp_divi} and \eqref{eq_beck}), naturally connects with the geodesic convexity of the functional $\mc{F}_{p,\sigma}$. 

Inspired by the result by Dietert \cite{dietert2015characterisation}, that if a Markov chain with finite states can be formulated as the gradient flow of the relative entropy, then it must be time-reversible, and its quantum analogs \cite{carlen2020non,cao2019gradient}, we consider the necessary condition for $\mc{P}_t^\dag$ being the gradient flow of $\mc{F}_{p,\sigma}$ for some Riemannian metric. Before we proceed, we give the following lemma that will be useful in the sequel.

\begin{lemma} \label{lem:key_rela}
For $\sigma \in \mathcal{D}_+(\mc{H})$, it holds that 
\begin{equation} \label{auxeqq_rela_00}
    [\sigma]_{p,0} = J_\sigma^{\kappa^{-1}_{1/p}} = R_\sigma \kappa_{1/p}^{-1}(\Delta_\sigma)\,,
   
\end{equation}
and 
\begin{align} \label{auxeqq_rela}
  [\sigma]_{p,\omega_j} \circ \partial_j \circ [\sigma]_{p,0}^{-1}   =   \Gamma_\si \circ \partial_j  \circ \Gamma_\si^{-1}\,,
\end{align}
where $[\sigma]_{p,0}$ and $[\sigma]_{p,\omega}$ are defined as in \eqref{def:kernel_multiplication}; the function $\kappa_{1/p}$ is the power difference \eqref{def:pdiff}. 
\end{lemma}

\begin{proof}
For the first identity \eqref{auxeqq_rela_00}, note from \eqref{eq:kernel_iden}, \eqref{eq:kernel_iden_2}, \eqref{auxeq_power_diff}, and definition \eqref{def:kernel_multiplication} that
\begin{equation*}
    [\sigma]_{p,0} = (\Omega_{\sigma}^{x^{-1}\varphi_p})^{-1} = R_\sigma \kappa_{1/p}^{-1}(\Delta_\sigma)\,.
\end{equation*}
With the help of the representation \eqref{eq:rep_gen} and the operator $[\cdot]_{p,\omega}$ defined in \eqref{def:kernel_multiplication}, we observe that \eqref{eq:rep_dbcl} can be reformulated as
\begin{align} \label{auxeqq_rela_0}
  - \langle X, \mc{L} X \rangle_{\sigma,\varphi_p} =  \sum_{j = J} \langle \Gamma_\si X, [\sigma]_{p,0}^{-1} \partial_j^\dag \Gamma_\si \partial_j X\rangle = \sum_{j = J}  \langle \Gamma_\si \partial_j X, [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X\rangle\,,
\end{align}
which is equivalent to the second identity \eqref{auxeqq_rela}.
\end{proof}

\begin{proposition}
If the dual primitive QMS $\mc{P}_t^\dag = e^{t \mc{L}^\dag}$ with $\sigma$-DBC is the gradient flow of the functional $\mc{F}_{p,\sigma}(\rho)$ with respect to some smooth Riemannian metric $g_\rho$, then $\mc{L}$ is self-adjoint with respect to the inner product:
\begin{equation} \label{def:nece_inner}
    \langle X , Y \rangle_{[\sigma]_{p,0}}: = \langle X, [\sigma]_{p,0} Y \rangle\,,\quad X, Y \in \mathcal{B}(\mc{H})\,,
\end{equation}
where $[\sigma]_{p,0}$ is defined by \eqref{def:kernel_multiplication}. 
\end{proposition}

\begin{proof}
Note that any Riemannian metric $g_\rho$ on $\mathcal{D}_+(\mc{H})$ can be written as 
\begin{align*}
    g_\rho(\nu_1,\nu_2) = \langle \mf{D}_\rho^{-1} \nu_1, \nu_2  \rangle\,,\quad \nu_1,\nu_2 \in T_\rho\,,
\end{align*}
where $\mf{D}_\rho$ is a positive definite operator from the  cotangent space $T^*_\rho$ to the tangent space $T_\rho$ at $\rho$ (both $T_\rho$ and $T^*_\rho$ can be identified with $\mc{B}^0_{sa}(\mc{H})$). The dual QMS $\mc{P}_t^\dag$ is the gradient flow of $\mc{F}_{p,\sigma}(\rho)$ with respect to $g_\rho$ means that 
\begin{align*}
    \mc{L}^\dag \rho = - \mf{D}_{\rho}(\delta_\rho \mc{F}_{p,\sigma}(\rho))\,.
\end{align*}
Substituting $\rho = \sigma + \varepsilon X$ for $X \in \mc{B}^0_{sa}(\mc{H})$ into the above formula and differentiating it with respect to $\varepsilon$ gives 
\begin{align*}
    \mc{L}^\dag X = - \mf{D}_{\rho} \Gamma_\si^{-1/\h{p}}f_p^{[1]}(\sigma^{1/p},\sigma^{1/p})\Gamma_\si^{-1/\h{p}}(X) = - \mf{D}_{\rho}[\sigma]^{-1}_{p,0} X\,,
\end{align*}
by \eqref{eq:deriv_func} and \eqref{eq:chain_curve}. Hence, it follows that, for $X, Y \in \mc{B}_{sa}^0(\mc{H})$, 
\begin{align*}
   \langle \mc{L}Y,  X\rangle_{[\sigma]_{p,0}} = - \langle Y, \mf{D}_\rho X \rangle = - \langle \mf{D}_\rho Y,  X \rangle = \langle Y, \mc{L} X \rangle_{[\sigma]_{p,0}}\,,
\end{align*}
which implies that $\mc{L}$ is self-adjoint on $\mc{B}(\mc{H})$ with respect to $\langle \cdot,\cdot \rangle_{[\sigma]_{p,0}}$, since $\mc{L}$ and $[\sigma]_{p,0}$ are Hermitian-preserving and satisfy
$\mc{L}({\bf 1}) = 0$ and $[\sigma]_{p,0}{\bf 1} = \sigma$. 
\end{proof}


In the limiting case $p \to 1^{+}$, corresponding to the gradient flow of
the Umegaki’s relative entropy, the inner product $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$ \eqref{def:nece_inner}  reduces to the BKM inner product:
\begin{equation*}
    \langle X, Y \rangle_{{\rm BKM}}: = \int_0^1 \tr(\sigma^{1-t} X^* \sigma^t Y)\ dt\,,
\end{equation*}
and hence our result  generalizes the one \cite[Theorem 2.9]{carlen2020non} by Carlen and Maas. In the case of $p = 2$, where $\mc{F}_{p,\sigma} $ is the variance (up to a constant factor), $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$ becomes the familiar KMS inner product $\langle \cdot,\cdot \rangle_{\sigma,1/2}$. The characterizations of, and the relations between the quantum Markov semigroups that are self-adjoint with respect to different inner products have been discussed extensively in \cite{carlen2020non,amorim2021complete,benoist2021deviation}. For our case, noting the formula \eqref{auxeqq_rela_00}, it follows from Lemma \ref{lem:self_adjoint} that the QMS satisfying $\sigma$-DBC is also self-adjoint with respect to the inner product $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$. But the reverse direction is still not well understood and needs further investigations.  





\subsection{Generalized quantum transport distances} \label{sec:quantum_distance} The Riemannian distance $W_{2,p}$ induced by the metric $g_{p,\rho}$ in \eqref{def:metric_tensor} can be defined as: for $\rho_0, \rho_1 \in \mathcal{D}_+(\mc{H})$, 
\begin{align} 
    W_{2,p}(\rho_0,\rho_1)^2  =\ & \inf  \left\{
    \int_0^1 g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right) ds\,;\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
    \right\}  \label{def:wp1} \\
     \overset{\eqref{def:gp2}}{=} & 
       \inf  \left\{
    \int_0^1 \norm{\nabla U(s)}^2_{p,\gamma(s)} ds \,;\ \dot{\gamma}(s) = \mf{D}_{p,\rho}U(s)\ \text{with}\  \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
    \right\}, \label{def:wp2}
\end{align}
where the infimum is taken over the smooth $(C^\infty)$ curves $\gamma(s):[0,1] \to \mathcal{D}_+(\mc{H})$. In this section, we will investigate the basic properties of the distance function $W_{2,p}$.  By the standard reparameterization techniques (cf.\cite[Lemma 1.1.4]{ambrosio2005gradient} or \cite[Theorem 5.4]{dolbeault2009new}),
we have that $W_{2,p}$ equals to the minimum arc length:
\begin{align} \label{def:wp3}
    W_{2,p}(\rho_0,\rho_1) = \inf  \left\{
    \int_0^1 g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right)^{1/2} ds\,;\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
    \right\},
\end{align}
where the infimum is taken over the smooth curves of constant speed (i.e., $g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right)^{1/2}$ is constant). Then it follows from the expression \eqref{def:wp3} that the Riemannian manifold $\mc{D}_+(\mc{H})$ equipped with the distance $W_{2,p}$ is a metric space. Moreover, it turns out that $W_{2,p}$ can be extended continuously to the boundary of $\mc{D}_+(\rho)$.

\begin{lemma} \label{lem:exten_bry}
The metric $W_{2,p}$ on $\mc{D}_+(\mc{H})$ extends continuously to $\mathcal{D}(\mc{H})$. 
\end{lemma}

Since the proof of Lemma \ref{lem:exten_bry} is similar to that of \cite[Proposition 9.2]{carlen2020non}, we sketch its main steps in Appendix \ref{app:proof} for completeness. In analogy with the classical 2-Wasserstein distance, it is helpful to introduce the quantum moment variable ${\bf B} = ([\rho]_{p,\omega_1}\partial_1 U, \ldots, [\rho]_{p,\omega_J}\partial_J U )$ for $U \in \mc{B}_{sa}^0$ and then reformulate \eqref{def:wp2} as a convex optimization problem: 
\begin{align} \label{def:wp4}
    W_{2,p}(\rho_0,\rho_1)^2 = \inf \Big\{
    \int_0^1 \norm{{\bf B}}^2_{-1,p,\gamma(s)} ds \,;\ &\dot{\gamma}(s) + {\rm div} {\bf B} (s) = 0\ \text{with}\  \gamma(0) = \rho_0\,, \gamma(1) = \rho_1,  \notag \\ 
    & \gamma(s) \in C\left([0,1]; \mathcal{D}(\mc{H})\right),\ \text{and}\  {\bf B} (s)\in L^1\left([0,1]; \mathcal{B}(\mc{H})^J\right) 
    \Big\}\,,
\end{align}
where the continuity equation $\dot{\gamma}(s) + {\rm div} {\bf B}(s) = 0$ is understood in the sense of distributions, and
$\norm{\cdot}_{-1,p,\rho}$ is the norm from 
the inner product $\langle\cdot,\cdot\rangle_{-1,p,\rho}$  on $\mc{B}(\mc{H})^J$ defined by
\begin{align*}
    \langle {\bf A}, {\bf B}\rangle_{-1,p,\rho} := \sum_{j = 1}^J \langle A_j, [\rho]_{p,\omega_j}^{-1} B_j\rangle\,.
\end{align*}
Indeed, by approximation techniques as in \cite[Proposition 1]{wirth2022dual} and a mollification argument from \cite[Lemma 2.9]{erbar2012ricci}, we can show that the infimum in the above representation \eqref{def:wp4} of $W_{2,p}$ can be equivalently taken over smooth curves $\gamma \in C^\infty([0,1];\mathcal{D}_+(\mc{H}))$. Then, the equivalence between the formulations \eqref{def:wp2} and \eqref{def:wp4} follows from the same arguments as in \cite[Lemma 9.1]{carlen2020non}, while the convexity of the optimization problem \eqref{def:wp4} is a simple consequence of the following lemma and \cite[Theorem 9.7]{carlen2020non}.





\begin{lemma}\label{lem:joint_convexity}
$\langle X, [\rho]_{p,\omega}^{-1}X \rangle$ is jointly convex in $(\rho,X) \in \mc{D}_+(\mc{H}) \times \mc{B}(\mc{H})$. 
\end{lemma}
\begin{proof}
Note from \eqref{def:kernel_multiplication} that  $$\langle X, [\rho]_{p,\omega}^{-1} X\rangle =  \langle \Gamma_\si^{-1/\h{p}}(X), f_p^{[1]}(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho),  e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho))(\Gamma_\si^{-1/\h{p}}(X))\rangle\,.$$ Then the statement follows from \cite[Lemma 2.3]{zhang2020optimal}, which shows the joint convexity of $\langle Y, f_p^{[1]}(A,B) Y \rangle$ for $Y \in \mathcal{B}(\mc{H})$ and full-rank $A,B \in \mc{B}_{sa}^+(\mc{H})$. 
\end{proof}

We summarize the above discussion as follows.








\begin{proposition} \label{prop:dist_convexity}
$W_{2,p}^2$ has an equivalent convex optimization formulation \eqref{def:wp4}. More precisely, let $\rho_0^i, \rho_1^i\in \mc{D}(\mc{H})$ for $i = 0,1$, and set $\rho_0^s := (1-s)\rho_0^0 + s\rho_0^1$ and $\rho_1^s := (1-s)\rho_1^0 + s\rho_1^1$ for $s \in [0,1]$. Then there holds 
\begin{equation*}
W_{2,p}(\rho^s_0,\rho^s_1)^2 \le (1-s) W_{2,p}(\rho^0_0,\rho^0_1)^2 + s W_{2,p}(\rho^1_0,\rho^1_1)^2\,.
\end{equation*}
\end{proposition}

The main result of this section is Theorem \ref{thm:main_wasser} below.

\begin{theorem}\label{thm:main_wasser}
$(\mc{D}(\mc{H}), W_{2,p})$ is a complete geodesic space. In particular, for any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$, the minimizer to \eqref{def:wp4} exists and gives the minimizing geodesic  $(\gamma_*(s))_{s \in [0,1]}$ connecting $\rho_0$ and $\rho_1$ and satisfying 
\begin{equation} \label{eq:geo_mini}
 W_{2,p}(\gamma_*(s),\gamma_*(t)) = |s - t| W_{2,p}(\rho_0,\rho_1)\,, \quad \forall s,t \in [0,1]\,.
\end{equation}
\end{theorem}
The completeness of the metric space $(\mc{D}(\mc{H}), W_{2,p})$
needs the following lemma, which is of interest itself. We postpone both proofs of Theorem \ref{thm:main_wasser} and Lemma \ref{lem:lower_bound} to Appendix \ref{app:proof} for ease of reading. 


\begin{lemma} \label{lem:lower_bound}
There exists a constant $C > 0$, independent of $p \in (1,2]$, such that for any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$,
\begin{align*}
    \norm{\rho_1 - \rho_0}_1 \le C W_{2,p}(\rho_0,\rho_1)\,.
\end{align*}
\end{lemma}


\begin{remark}
It is straightforward to generalize the arguments in \cite{wirth2022dual} to derive the dual formulation for the distance $W_{2,p}$ in terms of noncommutative Hamilton-Jacobi-Bellman-type equations. Actually, a formal calculation gives 
\begin{align*}
\frac{1}{2} W_{2,p}^2(\rho_0,\rho_1) = \sup\Big\{\tr(A(1)\rho_1 - A(0)\rho_0)\,;\ \tr(\dot{A}(t)\rho) + \frac{1}{2}\norm{\nabla A(t)}^2_{p,\rho} \le 0\,,\ \forall \rho \in \mathcal{D}(\mc{H}) \Big\}\,.
\end{align*}
The above dual formula allows us to fit our distance $W_{2,p}$ into the general framework of noncommutative transportation metrics proposed in \cite{gao2021ricci}, and to further discuss the coarse Ricci curvature of quantum channels with respect to $W_{2,p}$. The detailed investigation on these issues would be interesting but beyond the scope of this work. 
\end{remark}



We next derive the geodesic equations for the Riemannian manifold $(\mathcal{D}_+(\mc{H}), g_{p,\rho})$. Instead of regarding the geodesic equation as the Euler-Lagrange equation for the minimization problem \eqref{def:wp2} as in \cite{datta2020relating,carlen2020non}, we interpret it as the Hamiltonian flow of the Hamiltonian associated with the metric $g_{p,\rho}$: 
$$
H(\rho,U) := \frac{1}{2} g^{-1}_{p,\rho}(U,U) = \frac{1}{2}\langle \mf{D}_{p,\rho}U, U\rangle \quad \text{for}\  \rho \in \mathcal{D}_+(\mc{H})\ \text{and}\  U \in \mc{B}_{sa}^0(\mc{H})\,,
$$
where $g^{-1}_{p,\rho}$ is the inverse of the metric tensor. Then we can immediately write the geodesic equations:
\begin{align} \label{def:geo_eq}
&  \dot{\rho} = \delta_{U} H(\rho,U)\,,\quad   \dot{U} = -\delta_\rho H(\rho,U)\,.
\end{align}
By definition, it is clear that $\delta_{U} H(\rho,U) = \mf{D}_{p,\rho}U$. To find $\delta_\rho H$, by Lemma \ref{lem:high_order}, we have 
\begin{align} \label{eq:func_dev}
   \langle \delta_\rho H(\rho, U), A\rangle  =  & \lim_{\varepsilon \to 0} \frac{1}{2} \left\langle \mf{D}_{p, \rho + \varepsilon A} U, U\right\rangle \notag \\
    =  & \lim_{\varepsilon \to 0} \frac{1}{2} \left\langle \partial_j U, [\rho+\varepsilon A]_{p,\omega_j} \partial_j U \right\rangle \notag \\ 
     =  & \frac{1}{2} \sum_{j = 1}^J\left\langle \Gamma_\si^{1/\h{p}} (\partial_j U), (\delta_1 \theta_p)\big((l_j(\rho),l_j(\rho)), r_j(\rho)\big)\big(l_j(A), \Gamma_\si^{1/\h{p}}(\partial_j U)\big)\right\rangle  \notag 
    \\
        & +  \frac{1}{2} \sum_{j=1}^J \left\langle \Gamma_\si^{1/\h{p}} (\partial_j U), (\delta_2 \theta_p)\big(l_j(\rho), ( r_j(\rho), r_j(\rho))\big)\big(\Gamma_\si^{1/\h{p}}(\partial_j U), r_j(A)\big)\right\rangle\,, 
   
   
\end{align}
where for any $X \in \mathcal{B}(\mc{H})$, 
\begin{align*}
 l_j(X): = e^{\omega_j/2p} \Gamma_\si^{-1/\h{p}}(X)\,,\quad  r_j(X): = e^{-\omega_j/2p} \Gamma_\si^{-1/\h{p}}(X)\,.  
\end{align*}
We also note 
\begin{align} \label{eq:observation}
   & \left\langle \Gamma_\si^{1/\h{p}} (\partial_{j'} U), (\delta_1 \theta_p)\big((l_{j'}(\rho),l_{j'}(\rho)), r_{j'}(\rho)\big)\big(l_{j'}(A), \Gamma_\si^{1/\h{p}}(\partial_{j'} U)\big)\right\rangle  \notag \\
= &  \left\langle \Gamma_\si^{1/\h{p}} ((\partial_{j} U)^*), (\delta_1 \theta_p)\big((r_j(\rho),r_j(\rho)), l_j(\rho)\big)\big(r_j(A), \Gamma_\si^{1/\h{p}}((\partial_{j} U)^*)\big)\right\rangle \notag \\
= &  \left\langle \Gamma_\si^{1/\h{p}} (\partial_{j} U), (\delta_2 \theta_p)\big(l_j(\rho), (r_j(\rho),r_j(\rho))\big)\big(\Gamma_\si^{1/\h{p}}(\partial_{j} U), r_j(A)\big)\right\rangle \,,
\end{align}
where the first equality is from
the relations $l_j(\rho) = r_{j'}(\rho)$, $r_j(\rho) = l_{j'}(\rho)$, and $(\partial_j U)^* = - \partial_{j'}U$ by 
\eqref{eq:adjoint_index}, and the second equality is by the following formula from definitions \eqref{def_1:divi_diff} and \eqref{def_2:multiple}: 
\begin{equation*}
    \langle X^*, (\delta_1 f)((A,A),B)(Y,X^*) \rangle =  \langle X, (\delta_2 f)(B,(A,A))(X,Y)\rangle \,,
\end{equation*}
for any symmetric $f$: $f(s,t) = f(t,s)$, $X \in \mathcal{B}(\mc{H})$, $Y \in \mc{B}_{sa}(\mc{H})$, and commuting matrices $A,B \in \mc{B}_{sa}(\mc{H})$. The identity \eqref{eq:observation} implies that the two sums in \eqref{eq:func_dev} are actually equal. Therefore, we obtain from \eqref{def:geo_eq} and \eqref{eq:func_dev} the following proposition, where the local existence and uniqueness of geodesics follows from the standard theory of ODE. 

\begin{proposition}
On the Riemannian manifold $(\mathcal{D}_+(\mc{H}), g_{p,\rho})$, the unique constant speed geodesic $(\rho(t))_{t \in (-\varepsilon,\varepsilon)}$, $\varepsilon > 0$, with initial data: $\rho(0) = \rho_0 \in \mathcal{D}_+(\mc{H})$ and $\dot{\rho}(0) = \nu_0 \in \mc{B}_{sa}^0(\mc{H})$, satisfies the equation:
\begin{equation} \label{eq:geod_equa}
\begin{split}
    & \dot{\rho} = \mf{D}_{p,\rho}U \,,\\
    &   \langle \dot{U}, A\rangle  = - \sum_{j = 1}^J\left\langle \partial_j U,  \mc{K}_{\rho,A}^{(i),j} \left[ \partial_j U \right] \right \rangle\,, \quad \forall A \in \mc{B}^0_{sa}(\mc{H})\,,
\end{split}
\end{equation}
with $\rho(0) = \rho_0$ and $U(0) = \mf{D}_{p,\rho_0}^{-1} \nu_0$,
where $i = 1$ or $2$, and $\mc{K}_{\rho,A}^{(i),j}$ is defined by, for $\rho \in \mathcal{D}(\mc{H})$ and $A \in \mc{B}_{sa}^0(\mc{H})$, 
\begin{equation} \label{def:kgeode}
\begin{split}
 &\mc{K}_{\rho,A}^{(1),j}[\cdot] = \Gamma_\si^{1/\h{p}} \circ
 (\delta_1 \theta_p)\big((l_j(\rho),l_j(\rho)), r_j(\rho)\big)\big[l_j(A), \Gamma_\si^{1/\h{p}}(\cdot) \big]: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
 \\ & \mc{K}_{\rho,A}^{(2),j}[\cdot] = \Gamma_\si^{1/\h{p}} \circ (\delta_2 \theta_p)\big(l_j(\rho), ( r_j(\rho), r_j(\rho))\big)\big[\Gamma_\si^{1/\h{p}}(\cdot), r_j(A)\big]: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,.
\end{split}
\end{equation}
\end{proposition}


We end this section with the following proposition that the Riemannian metrics $g_{p,\rho}$ with $1 < p\le 2$ serve as an interpolating family between the metric defined by Carlen and Mass in \cite{carlen2017gradient} and the one induced from the KMS inner product $\langle \cdot, \cdot\rangle_{\sigma,1/2}$. It can be easily proved by an elementary analysis with
the fact, from the analytic perturbation theory \cite{rellich1969perturbation}, that the eigenvalues and eigenfunctions of $\Gamma_\si^{\scriptscriptstyle -1/\h{p}}(\rho)$ are differentiable with respect to $p$. 





\begin{proposition}\label{prop:repkernel}
Suppose that $\Gamma_\si^{-1/\h{p}}(\rho)$ has the eigen-decomposition:
$
    \Gamma_\si^{-1/\h{p}}(\rho) = \sum_{j = 1}^{d}  \lambda_{j,p} E_{j,p}. 
$ Then, we have 
\begin{align} \label{eq:repre_rpw}
    [\rho]_{p,\omega}A = \sum_{i,k = 1}^{d}
    \theta_p\left(e^{\omega/2p}\lambda_{k,p}, e^{-\omega/2p}\lambda_{i,p} \right)
   
    \Gamma_\si^{1/\h{p}} \left(E_{k,p} \Gamma_\si^{1/\h{p}}(A) E_{i,p}\right)\,, \quad A \in \mathcal{B}(\mc{H})\,,
\end{align}
where $\theta_p$ is given in \eqref{def:theta_p}. Moreover, $[\rho]_{p,\omega}$ is continuous in $p$, and it holds that when $p = 2$,
\begin{align*}
    [\rho]_{2,\omega}A = \Gamma_\si(A)\,,
\end{align*}
and when $p \to 1^+$, 
\begin{align} \label{def:metricpeq1}
     [\rho]_{p,\omega}A \to [\rho]_{\omega}A :=  \sum_{i,k = 1}^{d}
     \theta_{log}\left(e^{\omega/2}\lambda_k, e^{-\omega/2}\lambda_i \right)
   
     E_k A E_i\,,
\end{align}
where $\theta_{log}$ is the logarithmic mean:
\begin{align*}
    \theta_{log}(x,y): = \left\{
    \begin{aligned}
  & \frac{x - y}{\log x - \log y} \,, \quad & x \neq y\,, \\
   &  x \,,\quad  & x = y\,,
    \end{aligned}
    \right.
\end{align*}
and $\lambda_j$ are eigenvalues of $\rho$ with $E_j$ being the associated rank-one eigen-projections: $\rho = \sum_{j = 1}^{d} \lambda_j E_j$. 
\end{proposition}

Recalling the integral formula: $\frac{x - y}{\log x - \log y} = \int_0^1 x^{1-s} y^s \, ds $, we can write the operator $[\rho]_\omega$ in \eqref{def:metricpeq1} as
\begin{align*}
     [\rho]_\omega(A) = \int_0^1 e^{\omega(1/2-s)} \rho^{1-s} A \rho^{s}\, ds\,,\quad A \in \mathcal{B}(\mc{H})\,, 
   
\end{align*}
which is nothing but the noncommutative multiplication by $\rho$ involved in the definition of the quantum quadratic Wasserstein distance $W_2$ \cite{carlen2014analog,carlen2017gradient}. It follows that in the limiting case $p \to 1^+$, our transport distance $W_{2,p}$ reduces to the one $W_2$ introduced by Carlen and Maas. When $p = 2$, the representation \eqref{def:wp4} of $W_{2,p}$ gives
\begin{align} \label{eq:distance_1}
    W_{2,2}(\rho_0,\rho_1)^2 = \inf \Big\{
    \int_0^1 \sum_{j = 1}^J \langle B_j, \Gamma_\si^{-1} B_j  \rangle \, ds \,;\ &\dot{\gamma}(s) + {\rm div} {\bf B} (s) = 0\ \text{with}\  \gamma(0) = \rho_0\,, \gamma(1) = \rho_1 
    \Big\}\,,
\end{align}
which has a similar form as the classical distance $\mathscr{W}_{2,\alpha,\gamma}$ in \eqref{def:distance_savare} with $\alpha = 0$, and thus can be regarded as a quantum analog of dual $\dot{H}^{-1}$ Sobolev distance. These facts allow us to (at least formally) conclude that our new family of quantum distances $W_{2,p}$ interpolates the noncommutative $2$-Wasserstein and the $\dot{H}^{-1}$ Sobolev ones. One may further expect a stronger result that for $\rho_0,\rho_1 \in \mathcal{D}(\mc{H})$, $W_{2,p}(\rho_0,\rho_1)$ is continuous in $p \in (1,2]$ and $W_{2,p}(\rho_0,\rho_1) \to W_2(\rho_0,\rho_1)$ as $p \to 1^+$, but this task seems to be challenging, and we leave it for future investigation. 









\subsection{Ricci curvature and functional inequalities} \label{sec:geodesic_convexity} 
In this section, we will investigate the entropic Ricci curvature lower bound in terms of the quantum $p$-divergence $\mc{F}_{p,\sigma}(\rho)$. Thanks to the gradient flow structure obtained in Section \ref{sec:gradient}, we are allowed to derive some new functional inequalities from the Ricci curvature lower bound: a quantum HWI-type inequality and a transportation cost inequality, and connect them with the quantum Beckner's inequality \eqref{ineq_becp}, in the spirit of Otto and Villani \cite{otto2000generalization}. 


Let us first introduce the Ricci curvature in our setting. We follow the terminology of \cite{erbar2012ricci,datta2020relating}, and say that a primitive QMS $\mc{P}_t$ with $\sigma$-DBC has the Ricci curvature lower bound $\kappa \in \mathbb{R}$ associated with $\mc{F}_{p,\sigma}$ if
 \begin{equation} \label{eq:ricci_p}
    \frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s)) \ge \kappa g_{p,\gamma(0)}(\dot{\gamma}(0),\dot{\gamma}(0))\,,
\end{equation}
where $\gamma(s)$, $s \in (-\varepsilon,\varepsilon)$, is a geodesic satisfying $\gamma(0) = \rho \in \mathcal{D}_+(\mc{H})$. We compute the second derivative of $\mc{F}_{p,\sigma}$ along the constant-speed geodesic $\gamma(s)$. For this, noting the Riemannian gradient \eqref{eq:riemannian_grad} of $\mc{F}_{p,\sigma}$, by definition, we have 
\begin{align} \label{eq:first_dev_f}
     \frac{d}{ds} \mc{F}_{p,\sigma}(\gamma(s)) &= - g_{p,\gamma(s)} (\dot{\gamma}(s), \mc{L}^\dag(\gamma(s)))  = - \langle U(s), \mc{L}^\dag(\gamma(s))\rangle\,,
\end{align}
where $(\gamma,U)$ is the unique solution to \eqref{eq:geod_equa}. Then, differentiating \eqref{eq:first_dev_f} again with respect to $s$, we obtain  
\begin{align} \label{eq:second_dev_f}
     \frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s)) & = - \langle \dot{U}(s), \mc{L}^\dag(\gamma(s))\rangle - \langle U(s), \mc{L}^\dag(\dot{\gamma}(s))\rangle \Big|_{s = 0} \notag \\
     & = \sum_{j = 1}^J \big\langle \partial_j U(0),  \mc{K}_{\rho,\mc{L}^\dag(\gamma(0))}^{(i),j} \left[ \partial_j U(0) \right] \big \rangle - \left\langle U(0), \mc{L}^\dag(\mf{D}_{p,\gamma(0)} U(0)) \right \rangle\,.
\end{align}
Recall that the Riemannian Hessian of $\mc{F}_{p,\sigma}$ at $\rho \in \mathcal{D}_+(\mc{H})$ is defined by, for $U \in \mc{B}_{sa}^0(\mc{H})$, 
\begin{align*}
\hess \mc{F}_{p,\sigma}(\rho)[U,U] := \frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s))\,,
\end{align*}
where $(\gamma(s),U(s))$ satisfies the geodesic equation \eqref{eq:geod_equa} with initial conditions $\gamma(0) = \rho$, $U(0) = U$, and $\dot{\gamma}(0) = \mf{D}_{p,\rho}U$. We readily conclude from the formula \eqref{eq:second_dev_f} that 
\begin{align} \label{eq:hess_of_f}
    \hess \mc{F}_{p,\sigma}(\rho)[U,U] = \sum_{j = 1}^J \big\langle \partial_j U,  \mc{K}_{\rho,\mc{L}^\dag(\rho)}^{(i),j} \left[ \partial_j U \right] \big \rangle - \left\langle U, \mc{L}^\dag(\mf{D}_{p,\rho} U) \right \rangle\,, \quad \forall U \in \mc{B}_{sa}^0(\mc{H})\,.
\end{align}
Hence, it follows from definition that \eqref{eq:ricci_p} is equivalent to 
\begin{align} \label{eq:ricci_hess}
    \hess\mc{F}_{p,\sigma}(\rho)[U,U] \ge \kappa \langle U, \mf{D}_{p,\rho} U\rangle\,,  \tag{${\rm Ric}_p(\mc{L}) \ge \kappa$}
\end{align}
for $\rho \in \mathcal{D}_+(\mc{H})$ and $U \in \mc{B}^0_{sa}(\mc{H})$. The next proposition provides several equivalent characterizations of \eqref{eq:ricci_hess}, in terms of the $\kappa$-geodesic convexity of $\mc{F}_{p,\sigma}(\rho)$, the gradient estimates, and the evolution variational inequality. In what follows, we will use the notation:
\begin{align} \label{def:upper_deriv}
    \frac{d^+}{d t} f(t) = \limsup_{h \to 0^+} \frac{f(t+h) - f(t)}{h}\,.
\end{align}






\begin{proposition} \label{prop:geodesic_convex}
Let $\mc{P}_t$ be a primitive QMS satisfying $\sigma$-DBC with $\sigma \in \mathcal{D}_+(\mc{H})$. For $\kappa \in \mathbb{R}$, \eqref{eq:ricci_hess} is equivalent to the following statements:
\begin{enumerate}[(i)]
    \item $\mc{F}_{p,\sigma}(\rho)$ is geodesically $\kappa$-convex on $(\mc{D}(\mc{H}), W_{2,p})$, that is, for any constant-speed geodesic $(\gamma(s))_{s \in [0,1]} \subset \mathcal{D}(\mc{H})$, 
    \begin{align} \label{eq:geo_conv}
        \mc{F}_{p,\sigma}(\gamma(s)) \le (1-s) \mc{F}_{p,\sigma}(\gamma(0)) + s \mc{F}_{p,\sigma}(\gamma(1)) - \frac{\kappa}{2}s(1-s) W_{2,p}(\gamma(0),\gamma(1))^2\,.
    \end{align}
    \item For any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$, the following evolution
     variational inequality (EVI) holds: $\forall t \ge 0$,
     \begin{align} \label{eq:evi}
         \frac{1}{2}\frac{d^+}{d t} W_{2,p}(\mc{P}_t^\dag \rho_0 ,\rho_1)^2 + \frac{\kappa}{2} W_{2,p}(\mc{P}_t^\dag \rho_0 ,\rho_1)^2  \le \mc{F}_{p,\sigma}(\rho_1) - \mc{F}_{p,\sigma}(\mc{P}_t^\dag \rho_0)\,.
    \end{align}
\item The following gradient estimate holds: for $\rho \in \mathcal{D}_+(\mc{H})$ and $U \in \mc{B}_{sa}^0(\mc{H})$, $\forall t > 0$,
\begin{align*}
    \norm{\nabla \mc{P}_t(U)}^2_{p,\rho} \le e^{-2 \kappa t} \norm{\nabla U}^2_{p, \mc{P}_t^\dag\rho}\,.
\end{align*}
\item The following contraction of the transport distance $W_{2,p}$ along the gradient flow holds:
\begin{equation*}
  W_{2,p}(\mc{P}_t^\dag \rho_0, \mc{P}_t^\dag \rho_1) \le e^{-\kappa t} W_{2,p}(\rho_0,\rho_1) \quad \forall \rho_0,\rho_1 \in \mathcal{D}_+(\mc{H}) \,.
\end{equation*}
\end{enumerate}
\end{proposition}

\begin{proof}
The equivalence: \eqref{eq:ricci_hess} $\Longleftrightarrow (i) \Longleftrightarrow (ii)$ can be proved in the same manner as \cite[Theorem 3]{datta2020relating} with the gradient flow techniques from \cite{otto2005eulerian,daneri2008eulerian,dolbeault2012poincare}; see also \cite[Theorem 10.2]{carlen2020non}. \eqref{eq:ricci_hess} $\Longleftrightarrow (iii)$ follows from a similar semigroup interpolation argument as in the proof of \cite[Theorem 10.4]{carlen2020non}, while the proof of \eqref{eq:ricci_hess} $\Longleftrightarrow (iv)$ can be easily modified from those of 
Proposition 3.1 and (2.12) of \cite{daneri2008eulerian}. 
\end{proof}

With the notion of Ricci curvature, we next prove some interesting implications between functional inequalities, following the arguments of Otto and Villani \cite{otto2000generalization} (see also \cite{erbar2012ricci,datta2020relating}). 
 We start with an HWI-type inequality, which relates the generalized quantum transport distance $W_{2,p}$, the quantum $p$-divergence $\mc{F}_{p,\sigma}$, and the entropy production ($p$-Dirichlet form) $\mc{E}_{p,\mc{L}}$. The following lemma will be used later on.  
 
 
\begin{lemma} \label{lem:deriva_wasser}
Let $\rho, \w{\rho} \in \mathcal{D}_+(\mc{H})$ and $\rho_t = \mc{P}_t^\dag \rho$. Then there holds, for $t \ge 0$,
\begin{align*}
    \frac{d^+}{d t} W_{2,p}(\rho_t,\w{\rho}) \le  \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\,.
\end{align*}
\end{lemma}


\begin{proof}
By definition \eqref{def:upper_deriv} and the triangle inequality, we have 
\begin{align*} 
    \frac{d^+}{dt}W_{2,p}(\rho_t,\w{\rho}) = \limsup_{s \to 0^+} \frac{1}{s}\big(W_{2,p}(\rho_{t + s},\w{\rho}) - W_{2,p}(\rho_t,\w{\rho})\big) \le \limsup_{s \to 0^+} \frac{1}{s}W_{2,p}(\rho_t,\rho_{t+s})\,.
\end{align*}
The expression \eqref{def:wp3} with a time scaling gives 
\begin{align*}
W_{2,p}(\rho_t,\rho_{t+s}) \le \int_t^{t+s} \sqrt{g_{p,\gamma(\tau)}(\dot{\gamma}(\tau),\dot{\gamma}(\tau))} \ d\tau\,, 
\end{align*}
for any smooth curve $\gamma$ in $\mathcal{D}_+(\mc{H})$ such that $\gamma(t) = \rho_t$ and $\gamma(t+s) = \rho_{t+s}$.  Note from \eqref{eq:ep_divi} and \eqref{eq:grad_entropy} that 
\begin{align*}
    g_{p,\gamma(\tau)}(\dot{\gamma}(\tau),\dot{\gamma}(\tau)) = \frac{4}{p^2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\gamma(\tau)))\,.
\end{align*}
Therefore, we can find 
\begin{equation*}
    \frac{d^+}{dt}W_{2,p}(\rho_t,\w{\rho}) \le \limsup_{s \to 0^+} \frac{1}{s} \int_t^{t+s} \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\gamma(\tau)))}  \ d\tau = \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\,.   \qedhere
\end{equation*}
\end{proof}


\begin{theorem} \label{them:riccitohwi}
If \eqref{eq:ricci_hess} holds for some $\kappa \in \mathbb{R}$, then the following HWI-type inequality holds:
\begin{equation} \label{eq:HWI}
    \mc{F}_{p,\sigma}(\rho) \le \frac{2}{p} W_{2,p}(\rho, \sigma) \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))}- \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2\,, \quad \text{for all}\ \rho \in \mathcal{D}(\mc{H})\,. 
\end{equation}
\end{theorem}

\begin{proof}
It suffices to consider $\rho \in \mathcal{D}_+(\mc{H})$, since $\mathcal{D}_+(\mc{H})$ is dense in $\mathcal{D}(\mc{H})$ and the inequality \eqref{eq:HWI} is continuous with respect to $\rho$. Letting $\rho_t = \mc{P}_t^\dag \rho$ for $\rho \in \mathcal{D}_+(\mc{H})$, we derive from Lemma \ref{lem:deriva_wasser} that
\begin{align*}
    - \frac{1}{2} \frac{d^+}{d t}\Big|_{t =0} W_{2,p}(\rho_t, \sigma)^2 = &\liminf_{t \to 0^+} \frac{1}{2t} \big(W_{2,p}(\rho, \sigma)^2 - W_{2,p}(\rho_t, \sigma)^2\big) \\
    \le & \limsup_{t \to 0^+} \frac{1}{2t} \big( W_{2,p}(\rho, \rho_t)^2 + 2 W_{2,p}(\rho, \rho_t) W_{2,p}(\rho_t,\sigma) \big) \\
    \le &  \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))} W_{2,p}(\rho,\sigma)\,.
\end{align*}
Then, by above estimate, recalling the EVI \eqref{eq:evi} in Proposition \ref{prop:geodesic_convex} with $\rho_0 = \rho$ and $\rho_1 = \sigma$, we obtain
\begin{align*}
        \mc{F}_{p,\sigma}(\rho) & \le - \frac{1}{2}\frac{d^+}{d t}\Big|_{t= 0} W_{2,p}(\rho_t, \sigma)^2 - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2 \\
        & \le \frac{2}{p} W_{2,p}(\rho,\sigma) \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))} - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2 \,.  \qedhere
\end{align*}
\end{proof}

As a direct consequence of Theorem \ref{them:riccitohwi} above, in the case of positive Ricci curvature lower bound, we can obtain the quantum Beckner's inequality. 


\begin{corollary} \label{coro:poricci_beck}
If \eqref{eq:ricci_hess} holds for some $\kappa > 0$, then the quantum $p$-Beckner's inequality \eqref{eq_beck} holds with constant $\alpha_p \ge (\kappa p)/2 $. 
\end{corollary}

\begin{proof}
By Theorem \ref{them:riccitohwi} and Young's inequality, we have 
\begin{align*}
       \mc{F}_{p,\sigma}(\rho) \le \frac{2}{p} \Big( \frac{1}{2 C} W_{2,p}(\rho, \sigma)^2 + \frac{C}{2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho)) \Big) - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2\,.
\end{align*}
Letting $C = 2/(p\kappa)$, by definition \eqref{eq_beck}, we complete the proof. 
\end{proof}

Another implication of positive Ricci curvature is the finite diameter of the metric space $(\mathcal{D}(\mc{H}), W_{2,p})$, which can be viewed as a noncommutative Bonnet–Myers theorem.  



\begin{corollary}
If \eqref{eq:ricci_hess} holds for some $\kappa > 0$, then it holds that
\begin{equation} \label{est:diameter}
    \sup_{\rho_0, \rho_1\in \mc{D}(\mc{H})} W_{2,p}(\rho_0,\rho_1)^2 \le \frac{8}{\kappa p (p-1)} (\sigma_{\min}^{1-p} - 1)\,,
\end{equation}
where $\sigma_{\min}$ is the minimal eigenvalue of the invariant state $\sigma \in \mathcal{D}_+(\mc{H})$. 
\end{corollary}

\begin{proof}
Note that the geodesic convexity \eqref{eq:geo_conv} gives 
\begin{align} \label{auxest:diameter}
     \frac{\kappa}{8} W_{2,p}(\rho_0,\rho_1)^2 \le \frac{1}{2} \mc{F}_{p,\sigma}(\rho_0) + \frac{1}{2} \mc{F}_{p,\sigma}(\rho_1) \,.
\end{align}
Then the estimate \eqref{est:diameter} follows from \eqref{eq:rela_sandpdivi}, \eqref{eq:suprhodp}, and \eqref{auxest:diameter}.
\end{proof}


We say that a primitive QMS with $\sigma$-DBC satisfies a transport cost inequality associated with the distance $W_{2,p}$ with constant $c > 0$ if for all $\rho \in \mathcal{D}(\mc{H})$,
\begin{equation} \label{ineq:tc}
    W_{2,p}(\rho,\sigma) \le \sqrt{c \mc{F}_{p,\sigma}(\rho)}\,. \tag{TCp}
\end{equation}
We will show the chain of quantum functional inequalities \eqref{eq:chain}.




\begin{proposition} \label{propa}
Suppose that the quantum $p$-Beckner's inequality \eqref{eq_beck} holds for some $p \in (1,2]$. Then the transport cost inequality  \eqref{ineq:tc} holds with constant $c \ge p/\alpha_p$. 
\end{proposition}

\begin{proof}
Again, it suffices to consider $\rho \in \mathcal{D}_+(\mc{H})$. Let $\rho_t = \mc{P}_t^\dag \rho$ and define the function 
\begin{align*}
    h(t) := W_{2,p}(\rho_t,\rho) + \sqrt{c \mc{F}_{p,\sigma}(\rho_t)}\,, \quad  t \ge 0\,.
\end{align*}
Clearly, $h(t)$ satisfies that $h(0)= \sqrt{c \mc{F}_{p,\sigma}(\rho)}$ and $h(t) \to W_{2,p}(\sigma,\rho)$ as $t \to \infty$ by \eqref{eq:conver_qms}. We now claim that when $c \ge p/\alpha_p$, $\frac{d^+}{d t} h(t) \le 0$ holds for $t \ge 0$, which completes the proof. By Lemma \ref{lem:deriva_wasser} and \eqref{eq:ep_divi}, when $\rho_t \neq \sigma$, we compute 
\begin{align*}
    \frac{d^+}{d t} h(t) &\le \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))} -  \frac{2\sqrt{c}}{p^2 \sqrt{\mc{F}_{p,\sigma}(\rho_t)}} \mc{E}_{p,\mc{L}}\big(\Gamma_\si^{-1}(\rho_t)\big) \\
    & = \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))} \Big(1 -  \frac{\sqrt{c}}{p \sqrt{\mc{F}_{p,\sigma}(\rho_t)}} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\big) \Big) \le 0\,,
\end{align*}
where the last inequality follows from $c \ge p/\alpha_p$ and the Beckner's inequality \eqref{eq_beck}. If $\rho_{t_0} = \sigma$ for some $t_0$, then $\rho_t = \sigma$ for $t \ge t_0$ and hence $\frac{d^+}{d t} h(t) = 0$ for $t \ge t_0$. 
\end{proof}

\begin{proposition} \label{propb}
If the transport cost inequality \eqref{ineq:tc} holds with constant $c$, then the Poincar\'{e} inequality \eqref{ineq_pi} holds with $f = \varphi_p$ and constant $\lambda \ge 2/c$.
\end{proposition}

\begin{proof}
We consider $X \in \mc{B}_{sa}(\mc{H})$ with $\tr(\sigma X) = 0$, and define $\rho_\varepsilon = \Gamma_\si({\bf 1} + \varepsilon X)$.  Recall Theorem \ref{thm:main_wasser} and let 
$(\gamma_\varepsilon, {\bf B}_\varepsilon)$ be the minimizer to \eqref{def:wp4} for $W_{2,p}(\rho_\varepsilon,\sigma)$. Then, note from \eqref{eq:chi_heisen} that 
\begin{align} \label{auxeqq_tctopi}
    \norm{X}_{\sigma,\varphi_p}^2 & =  \frac{1}{\varepsilon^2} \Big\langle \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma), \int_0^1 - {\rm div} {\bf B}_\varepsilon(s)  ds    \Big\rangle \notag \\
    & \le \frac{1}{\varepsilon^2} \Big( \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds  \Big)^{1/2} \Big(  \int_0^1 \norm{{\bf B}_\varepsilon(s)}^2_{-1,p,\gamma_\varepsilon(s)}  ds  \Big)^{1/2} \notag \\
     & \le \frac{1}{\varepsilon^2} \Big( \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds  \Big)^{1/2} W_{2,p}(\rho_\varepsilon,\sigma)\,,
\end{align}
by using the continuity equation in the first line, and Cauchy's inequality in the second line. Applying \eqref{ineq:tc} with the expansion \eqref{eq:asy_divi}, by definition \eqref{def:quanpdivi} of $\mc{F}_{p,\sigma}(\rho)$ and the relation \eqref{eq:chi_heisen}, 
we find 
\begin{align} \label{auxeq:wasserexp}
    \frac{1}{\varepsilon} W_{2,p}(\rho_\varepsilon,\sigma) \le \sqrt{ \frac{c}{2} \norm{X}_{\sigma,\varphi_p}^2 + O(\varepsilon)}\,.
\end{align}
A direct computation with \eqref{eq:kernel_iden_2} and Lemma \ref{lem:key_rela} gives  
\begin{align} \label{newlabel_aux}
    \partial_j \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma) & = \varepsilon \partial_j \Gamma_\si^{-1}\varphi_p(\Delta_\sigma) R_\sigma \Gamma_\si^{-1} \Gamma_\si X \notag \\ 
    & = \varepsilon \partial_j [\sigma]_{p,0}^{-1} \Gamma_\si X  \notag\\
    & = \varepsilon [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j  X\,.
\end{align}
By the proof of Lemma \ref{lem:exten_bry}, $\norm{\rho_\varepsilon -\sigma}_1 \to 0$ as $\varepsilon \to 0$ implies $W_{2,p}(\rho_\varepsilon,\sigma) \to 0$, which further yields $W_{2,p}(\gamma_\varepsilon(t),\sigma) = |1 - t| W_{2,p}(\rho_\varepsilon,\sigma) \to 0$, as $\varepsilon \to 0$, for all $t \in (0,1)$. Moreover, using Lemma \ref{lem:lower_bound}, we have  $\norm{\gamma_\varepsilon(t)-\sigma}_1  \to 0$, as $\varepsilon \to 0$, for  $t \in (0,1)$. 
It follows from the dominated convergence theorem that
\begin{align*}
   \frac{1}{\varepsilon^2} \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds  & \overset{\eqref{newlabel_aux}}{=} \sum_{j = 1}^J \int_0^1 \langle [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j  X, [\gamma_\varepsilon(s)]_{p,\omega_j} [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j  X \rangle \\
   & \to  \sum_{j = 1}^J \int_0^1 \langle  \Gamma_\si \partial_j  X,  [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j  X \rangle \overset{\eqref{auxeqq_rela_0}}{=}  - \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,, \quad \text{as}\ \varepsilon \to 0\,.
\end{align*}
 Combining the above formula with \eqref{auxeqq_tctopi} and \eqref{auxeq:wasserexp}, we conclude 
\begin{equation*}
     \norm{X}_{\sigma,\varphi_p}^2 \le - \frac{c}{2}  \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,. \qedhere
\end{equation*}
\end{proof}

We have seen that the entropic Ricci curvature lower bound \eqref{eq:ricci_hess} can imply a sequence of quantum functional inequalities. A natural and important following-up question is how to estimate the lower bound $\kappa$ for the Ricci curvature. Following closely the arguments in \cite[Theorem 10.6]{carlen2020non}, we can explicitly estimate the Ricci curvature lower bound for the depolarizing semigroup by definition \eqref{eq:ricci_hess}. 

\begin{proposition}  \label{prop:ricci_depol}
Let $\mc{L}_{depol}$ be the generator \eqref{def:depol} for the depolarizing semigroup with $\gamma > 0$ and the invariant state $\sigma = {\bf 1}/d$. Then the Ricci curvature of $\mc{L}_{depol}$ is bounded below by $\gamma p/2$.
\end{proposition}


\begin{proof}
Note from the definition of $\mc{L}_{depol}$ that $\partial_j \mc{L}_{depol} = - \gamma \partial_j$.
Recalling \eqref{eq:hess_of_f}, and we compute 
\begin{align*}
    -\langle \mc{L}_{depol} U, \mf{D}_{p,\sigma} U \rangle =  \gamma d^{1-p} \sum_{j = 1}^J  \langle \partial_j U, \theta_p(\rho,\rho)   \partial_j U\rangle\,.
\end{align*}
By definition \eqref{def:kgeode}, we can also calculate 
\begin{align*}
    \sum_{j = 1}^J \big\langle \partial_j U,  \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j} \left[ \partial_j U \right] \big \rangle & = \gamma d^{1-p} \sum_{j =1}^J \big\langle  \partial_j U,
 (\delta_1 \theta_p)\big((\rho,\rho), \rho\big)\big[ \frac{{\bf 1}}{d} - \rho, \partial_j U \big] \big\rangle \\
 & =  \gamma d^{1-p} \sum_{j =1}^J \big\langle  \partial_j U, L_{\frac{{\bf 1}}{d} - \rho} (\partial_1 \theta_p)(\rho,\rho)[\partial_j U] \big\rangle\,,
\end{align*}
since $\mc{L}^\dag_{depol} = \gamma\big(\frac{{\bf 1}}{d} - \rho \big)$. Similarly, we have 
\begin{align*}
    \sum_{j = 1}^J \big\langle \partial_j U,  \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(2),j} \left[ \partial_j U \right] \big \rangle 
 =  \gamma d^{1-p} \sum_{j =1}^J \big\langle  \partial_j U, R_{\frac{{\bf 1}}{d} - \rho} (\partial_2 \theta_p)(\rho,\rho)[\partial_j U] \big\rangle\,.
\end{align*}
It follows that 
\begin{align*}
      \hess \mc{F}_{p,\sigma}(\rho)[U,U] &= \frac{1}{2} \sum_{j = 1}^J \Big\langle \partial_j U,  \big(\mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j} + \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j}\big) \left[ \partial_j U \right] \Big \rangle - \left\langle U, \mc{L}_{depol}^\dag(\mf{D}_{p,\rho} U) \right \rangle \\
      & =  \gamma d^{1-p} \sum_{j =1}^J \Big\langle  \partial_j U, \big(\frac{1}{2d}\partial_1 \theta_p + \frac{1}{2d}\partial_2 \theta_p + \frac{p}{2}\theta_p \big)     (\rho,\rho)[\partial_j U] \Big\rangle \\
      & \ge d^{1-p} \frac{p\gamma }{2} \sum_{j =1}^J \Big\langle  \partial_j U, \theta_p     (\rho,\rho)[\partial_j U] \Big\rangle = \frac{p\gamma}{2} \langle U, \mf{D}_{p,\rho} U\rangle\,,
\end{align*}
where the second line is from $ x \partial_x \theta_p(x,y) +  y \partial_y \theta_p(x,y) = (2-p) \theta_p(x,y)$; the third inequality follows from $ \partial_x \theta_p(x,y) +  \partial_y \theta_p(x,y) \ge 0$ by the concavity of $x^{p-1}$.
\end{proof}

However, similarly to the case of estimating the functional inequality constant, there are very few examples where the explicit expressions of the Ricci curvature lower bounds can be obtained. To avoid the complicated computation and estimation based on the definition \eqref{eq:ricci_hess}, Carlen and Maas \cite{carlen2017gradient} consider the following intertwining property of the quantum Markov
semigroup: for some $\kappa \in \mathbb{R}$ and all $j$, 
\begin{align} \label{eq:intertwin}
    \partial_j \mc{P}_t = e^{- \kappa t} \mc{P}_t \partial_j\,,
\end{align}
which can be verified for many interesting cases, e.g., Fermi and Bose Ornstein-Uhlenbeck semigroups \cite[Section 6]{carlen2017gradient}. They showed that under the condition \eqref{eq:intertwin}, the Ricci curvature of the QMS $\mc{P}_t$ associated with the Umegaki's relative entropy is bounded from below by $\kappa$. The key step in their argument is the monotonicity of the action functional:  
\begin{align} \label{eq:monotone}
    \langle \mc{P}^\dag_t A, [\mc{P}^\dag_t\rho]_{\omega}^{-1} \mc{P}^\dag_t A\rangle \le  \langle A, [ \rho]_{\omega}^{-1}   A\rangle\,, 
\end{align}
where $\mc{P}_t$ is the primitive QMS satisfying $\sigma$-DBC. To extend their approach to our case, we need to show that a similar monotonicity result as \eqref{eq:monotone} holds for $ \langle A, [\rho]_{p,\omega}^{-1}  A\rangle$. It is a nontrivial task, since $\langle A, [\rho]_{p,\omega}^{-1}  A\rangle$ is not $1$-homogeneous so that the contractivity of $\langle A, [\rho]_{p,\omega}^{-1}  A\rangle$ under $\mc{P}_t^{\scriptscriptstyle \dag}$ can not be implied from its joint convexity \cite{lesniewski1999monotone}. For the symmetric QMS, $\mc{P}_t = \mc{P}_t^{\scriptscriptstyle \dag}$ is the unital quantum channel for each $t$. Note that \cite[Theorem 5.1]{zhang2021some} has shown that 
\begin{equation} \label{auxeq_p-nomo}
       \langle \Phi(A), [\Phi(\rho)]_{p,0}^{-1} \Phi(A)\rangle \le  \langle A, [ \rho]_{p,0}^{-1}  A\rangle\,, \quad \forall \rho \in \mathcal{D}_+(\mc{H})\,, A \in \mathcal{B}(\mc{H})\,,
\end{equation}
holds for any unital quantum channel $\Phi$. Thus, in this case, it is straightforward to conclude as in \cite[Theorem 10.9]{carlen2020non} that if the primitive symmetric QMS $\mc{P}_t$ satisfies 
the property \eqref{eq:intertwin},  then its Ricci curvature associated with $\mc{F}_{p,\sigma}$ has a lower bound $\kappa$. It is known \cite{carlen2017gradient,carlen2020non} that the infinite temperature Fermi Ornstein-Uhlenbeck semigroup is symmetric and satisfies \eqref{eq:intertwin} with $\kappa = 1$, which readily gives that it has Ricci curvature lower bound $1$ and then the quantum Beckner's inequality holds with $\alpha_p \ge p/2$ by Corollary \ref{coro:poricci_beck}. But it seems not easy to extend the monotonicity result \eqref{auxeq_p-nomo} beyond the symmetric regime, namely, to show the monotonicity $\langle A, [\rho]_{p,\omega}^{-1}  A\rangle$ under quantum channels with $\sigma$-DBC. We choose to investigate it in the future. 



\section{Conclusions and discussion}
In this work, we have introduced the families of quantum Beckner's inequalities \eqref{ineq_becp} and \eqref{ineq_dbecq} that interpolate between the Sobolev-type and Poincar\'{e} inequalities. The basic properties of Beckner's inequalities, e.g., the monotonicity, the uniform
positivity, and the stability of the optimal constants, have been investigated in detail. We have also discussed their relations with other known quantum functional inequalities and applied Beckner's inequalities \eqref{ineq_becp} to estimating the mixing time and deriving the moment estimates. Furthermore, we have provided a quantum optimal transport framework for analyzing the Beckner's inequalities. In doing so, we have defined a new class of quantum transport distances $W_{2,p}$ such that the QMS with $\sigma$-DBC is the gradient flow of the quantum $p$-divergence. The main properties of the metric space $(\mathcal{D}(\mc{H}),W_{2,p})$ have been analyzed. 
We have then introduced the associated entropic Ricci curvature and showed that it could yield a number of implications between \eqref{ineq_becp}, \eqref{ineq_pi}, an HWI-type inequality, and a transport cost inequality. 

We briefly discuss some generalizations and applications of our results and methods. First, as mentioned in the introduction, the tensorization property for quantum functional inequalities is much more subtle than the classical case. Some tensorization-type results have been obtained for the quantum MLSI and LSI in the case of the depolarizing semigroups, e.g., \cite[Lemma 25]{kastoryano2013quantum} and \cite[Theorems 19 and 21]{beigi2020quantum}. By Lemma \ref{lem:aux_func} and Corollary \ref{coro:pitobeck}, these results can also provide dimension-independent lower bounds for the quantum Beckner constants (in certain scenarios). Second, thanks to Lemma \ref{lem:two-sided}, we can estimate the strong data processing inequality constant for the quantum $p$-divergence $\mc{F}_{p,\sigma}$ in terms of the $\chi^2_{\kappa_{1/p}}$ contraction coefficient, in the sense of \cite[Theorem 4.1]{gao2021complete}. We can also discuss the approximate tensorization property of $\mc{F}_{p,\sigma}$ similarly to \cite[Theorem 5.1]{gao2021complete} and the stability of the data processing inequality for the divergence $\mc{F}_{p,\sigma}$ similarly to \cite[Proposition 5.1]{junge2019stability}. 
Third, in view of \cite{wirth2021curvature}, it is straightforward to consider the curvature-dimension conditions for quantum systems and investigate the finite-dimension version of the quantum Beckner's inequality. The details and refinements of these results would be worth being reported elsewhere.

We conclude with some interesting open questions. First, although we have obtained several generic results for the quantum Beckner constant, it is also desirable to establish more quantitative estimates for some concrete, physically meaningful models, for instance, the quantum spin system \cite{capel2020modified} and the quantum Markov semigroups constructed from the classical ones via the group transference \cite{gao2020fisher}. Second, the Bakry-\'{E}mery method has been well explored in the diffusion and discrete cases for proving Beckner's inequalities \cite{jungel2017discrete,weber2021entropy,gentil2021family}. However, the quantum extension of such an approach is still unknown. We are also curious about whether there exists some connections between Beckner's inequalities and the hypercontractivity of QMS, considering the existing works \cite{beckner1989generalized,arnold2007interpolation} and 
the natural relations between the Beckner's inequality and the noncommutative $p$-norm $\norm{\cdot}_{p,\sigma}$. Third, in this work we have only considered the quantum Beckner's inequality for the primitive QMS. It would be important and useful to extend it to the non-primitive setting. The study of non-primitive quantum functional inequalities is initiated by Bardet \cite{bardet2017estimating} and has become an active research topic in recent years. It is known from \cite{frigerio1982long,gao2021complete} that for a general QMS, we have $E_\mc{F} \mc{P}_t = \mc{P}_t E_\mc{F} = E_\mc{F}$ and 
\begin{align*}
\lim_{t \to \infty} \mc{P}_t (X -  E_{\mc{F}}(X)) = \lim_{t \to \infty} \mc{P}_t X -  E_{\mc{F}}(X) = 0\,,
\end{align*}
where $E_\mc{F}$ is the conditional expectation to the  fixed-point algebra $\mc{F} := \left\{X \in \mathcal{B}(\mc{H});\ \forall t \ge 0,\, \mc{P}_t(X) = X \right\}$. One may attempt to define the non-primitive Beckner's inequality as 
\begin{equation*}
     \alpha_p \Big(\norm{X}^p_{p,\sigma} - \norm{E_\mc{F}(X)}_{p,\sigma}^p \Big) \le (p-1) \mc{E}_{p,\mc{L}}(X)\,, \quad \forall X \ge 0\,,
\end{equation*}
where $\sigma \in \mc{F}$ is a reference invariant state. 
However, to make it well-defined, we need to show that the optimal constant $\alpha_p(\mc{L})$ is independent of the choice of $\sigma$, or we consider a particular state $\sigma_{\tr} = E_{\mc{F}}^\dag(\frac{{\bf 1}}{d})$ as in \cite{bardet2017estimating}. Alternatively, we may define the inequality by \eqref{eq_beck} with $\sigma = E_\mc{F}^\dag(\rho)$, i.e., consider the divergence $\mc{F}_{p,E_\mc{F}^\dag(\rho)}(\rho)$. 
It is unclear which one is the most appropriate definition. It would be also interesting to further consider the complete Beckner's inequality and its properties in the spirit of \cite{gao2020fisher,brannan2021complete,brannan2022complete,gao2021complete}.


























\begin{appendix}

\section{Some additional preliminaries}

\subsection{Quantum \texorpdfstring{$\chi^2$}--divergence} \label{app:quantum_diver} 

 In this appendix, we briefly recall the quantum $\chi^2$-divergences introduced in \cite{temme2010chi}. We consider the following class of functions:  
\begin{align*}
     \mc{K} = \{\kappa: (0,\infty) \to (0,\infty)\,;\ \kappa \ \text{is operator convex},\ x\kappa(x) = \kappa(x^{-1}),\ \kappa(1) = 1\}.
\end{align*}
By \cite[Proposition 2.1]{hiai2016contraction}, for $\kappa \in \mc{K}$, there exists a unique probability measure $m$ on $[0,1]$ such that 
\begin{align} \label{eq:inte_rep}
    \kappa(x) = \int_0^1 \left(\frac{1}{x + s} + \frac{1}{sx + 1}\right) \frac{1 + s}{2} dm\,,
\end{align} 
which implies 
\begin{align} \label{eq:compare_function}
    \frac{2}{1 + x} \le \kappa(x) \le \frac{1+x}{2 x}\,,\quad  \forall \kappa \in \mc{K}\,.
\end{align}
We now define the operator $\Omega_\sigma^\kappa$ for $\kappa \in \mc{K}$ and $\sigma > 0$:
\begin{align} \label{def:omega}
    \Omega_\sigma^\kappa = R_\sigma^{-1} \kappa(\Delta_\sigma):\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,.
\end{align}
Recalling $J_\sigma^f$ in \eqref{def:operator_kernel}, clearly there holds
\begin{align} \label{aux_rela}
    (\Omega_\sigma^\kappa)^{-1} = J_\sigma^{1/\kappa}\,.
\end{align}
\begin{definition} \label{def:quantumchi}
The quantum $\chi_\kappa^2$-divergence for $\rho, \sigma \in \mathcal{D}(\mc{H})$ is defined by
\begin{align*}
    \chi_{\kappa}^2(\rho,\sigma) = \langle \rho - \sigma, \Omega_\sigma^\kappa(\rho-\sigma)\rangle\,,
\end{align*}
when ${\rm supp}(\rho) \subset {\rm supp}(\sigma)$; otherwise $\chi_\kappa^2(\rho,\sigma) = \infty$.  
\end{definition}

For the purposes of this work, the following family of power difference means is of particular interest \cite{hiai1999means}:
\begin{align}\label{def:pdiff}
\kappa_\alpha = \frac{\alpha}{\alpha-1} \frac{x^{\alpha - 1} - 1}{x^{\alpha} - 1}\,, \quad \alpha \in [-1,2]\,.
\end{align}
In fact, the kernel function of the operator $J_\sigma^{1/k}$ in \eqref{aux_rela}  is given by  
\begin{align*}
    M_\alpha = y \kappa_\alpha^{-1}(x/y) = \frac{\alpha - 1}{\alpha}\frac{x^\alpha - y^\alpha}{x^{\alpha - 1} - y^{\alpha - 1}}\,,
\end{align*} 
which is the so-called A-L-G  interpolation mean since $M_\alpha$, for $\alpha = -1$, $\alpha = 1/2$, $\alpha  = 1$, and $\alpha = 2$, gives the harmonic mean,  the geometric mean, the logarithmic mean, and the arithmetic mean, respectively. 











\subsection{Noncommutative calculus} \label{app:NCC}
In this appendix, following \cite{carlen2020non,bardet2017estimating}, we 
review some fundamentals about noncommutative calculus associated with the derivations $\partial_j$. Let operators $A,B \in \mc{B}_{sa}(\mc{H})$ admit the spectral decompositions:
\begin{align*}
    A = \sum_{i = 1}^{d}\lambda_i A_i\,,\quad B = \sum_{k=1}^{d} \mu_k B_k\,,
\end{align*} 
where $\lambda_i$ and $\mu_k$ are eigenvalues of $A$ and $B$, respectively; $A_i$ and $B_i$ are the associated rank-one spectral projections. For a function $f \in C(I \times I)$ with $I$ being a compact interval containing the spectra of $A$ and $B$,  we define the Schur multiplier (double sum operator) by \cite{birman2003double,potapov2011operator,de2004differentiation}
\begin{align} \label{def:doutble_op_sum}
    f(A,B) = \sum_{i,k = 1}^{d} f(\lambda_i,\mu_k) L_{A_i}R_{B_k}\,,
\end{align}
where $C(I \times I)$ is the Banach space of complex-valued continuous functions on $I \times I$. It was observed in \cite{bardet2017estimating} that given $A,B \in \mc{B}_{sa}(\mc{H})$, $f(A,B)$ is $*$-representation between $C(I \times I)$ and $\mc{B}(\mc{B}(\mc{H}))$. Indeed, we have the following lemma from \cite[Lemma 4.1]{bardet2017estimating} and \cite[Lemma 6.6]{carlen2020non}. 
\begin{lemma} \label{lem:double_inner}
Let $A,B \in \mc{B}_{sa}(\mc{H})$ and the compact interval $I$ contain the spectra of $A$ and $B$. It holds that 
\begin{enumerate}[1.]
    \item $f(A,B)g(A,B) = (fg)(A,B)$ for $f,g \in C(I \times I)$. 
    \item If $f \in C(I \times I)$ is non-negative, then $f(A,B)$ is a positive semidefinite operator on $B(\mc{H})$ with respect to the inner product $\langle \cdot, \cdot \rangle$. It follows that if $f$ is strictly positive, the sequilinear form $\langle \cdot, f(A,B)(\cdot)\rangle$ defines an inner product on $\mc{B}(\mc{H})$. 
\end{enumerate}
\end{lemma}





In this work, we mainly consider the case where $f$ is the divided difference of some differentiable function $\varphi$ on $I$:
\begin{align} \label{def:divi_diff}
    \varphi^{[1]}(\lambda,\mu) = \left\{
    \begin{aligned}
  &  \frac{\varphi(\lambda)-\varphi(\mu)}{\lambda-\mu}\,, \quad & \lambda \neq \mu\,, \\
   & \varphi'(\lambda)\,,\quad  & \lambda = \mu\,,
    \end{aligned}
    \right.
\end{align}
which is closely related to the chain rule for the derivation $\partial_j$ (cf.\cite[Lemma 4.2]{bardet2017estimating} and \cite[Proposition 6.2]{carlen2020non}). 
\begin{lemma} \label{lem:chain_rule}
Under the same assumption as in Lemma \ref{lem:double_inner}, for a continuously differentiable function $f \in C(I)$ such that $f(0) = 0$, we have, for any $V \in \mathcal{B}(\mc{H})$,
\begin{align} \label{eq:chain_rule}
    V f(B) - f(A) V = f^{[1]}(A,B)(V B - A V)\,.
\end{align}
Moreover, in the case of $A = B$, the identity  \eqref{eq:chain_rule} holds
for any $f: I \to \mathbb{C}$ (that may not be differentiable).
\end{lemma}
It then follows from the above lemma that for a differentiable curve $A(t): (a,b) \to \mc{B}(\mc{H})$, we have 
\begin{align}\label{eq:chain_curve}
& \frac{d}{d t} f(A(t)) = f^{[1]}(A(t),A(t))(A'(t))\,.
\end{align}
We also need a multiple operator version of \eqref{eq:chain_curve}. We recall that for a differentiable function $\varphi: \mathbb{R}^n \to \mathbb{C}$, the partial divided difference $\delta_j \varphi: \mathbb{R}^{n+1} \to \mathbb{C} $ with respect to the variable 
$x_j$ is defined by 
\begin{align} \label{def_1:divi_diff}
(\delta_{j} \varphi)(x_1,\cdots,x_{j-1},(\lambda,\mu),x_{j+1},\cdots,x_n)=  \left(\varphi(x_1,\cdots,x_{j-1},\cdot,x_{j+1},\cdots,x_n)\right)^{[1]}(\lambda,\mu).
\end{align}
Let $A^{(k)}$, $k = 1,\ldots,n$, be self-adjoint operators with the spectral decompositions: $A_i = \sum_{i} \lambda_{i}^{(k)}A_{i}^{(k)}$, where $\lambda_i^{(k)}$ are eigenvalues and $A_{i}^{(k)}$ are the associated rank-one spectral projections. For a function $\varphi: I \times \cdots \times I \to \mathbb{C}$ with the interval $I$ containing the spectra of $A^{(i)}$,
the multiple operator sum is defined as:
\begin{align} \label{def_2:multiple}
    \varphi(A_1,\cdots,A_n) = \sum_{i_1,\cdots,i_n = 1}^{d} \varphi(\lambda^{(1)}_{i_1},\cdots,\lambda^{(n)}_{i_n})A_{i_1}^{(1)}\otimes\cdots \otimes A_{i_n}^{(n)}.
\end{align}
The following chain rule from \cite[Proposition 6.8]{carlen2020non} shall be useful in the expression of the geodesic equations for the generalized quantum transport distance.
\begin{lemma} \label{lem:high_order}
Let the curves $A_t, B_t: (a,b) \to \mc{B}_{sa}(\mc{H})$ be differentiable, and let $\varphi: I \times I \to \mathbb{C}$ be differentiable with $I$ containing the spectra of $A_t$ and $B_t$ for all $t \in (a,b)$. Then there holds 
\begin{align*}
   \partial_t \varphi(A_t,B_t)(\cdot) = (\delta_1 \varphi) ((A_t,A_t),B_t)[\partial_t A_t, \cdot] + (\delta_2 \varphi)(A_t,(B_t,B_t))[\cdot, \partial_t B_t].
\end{align*}
\end{lemma}





\section{Additional Proofs} \label{app:proof}
\begin{proof}[Proof of Lemma \ref{lem:exten_bry}]
Consider $\rho_0,\rho_1 \in \mathcal{D}(\mc{H})$ and let $\{\rho_0^n\}, \{\rho_1^n\}$ be any sequences in $\mathcal{D}_+(\mc{H})$ such that $\norm{\rho_i^n - \rho_i}_2 \to 0$ as $n \to \infty$ for $i = 0,1$. It suffices to show that $W_{2,p}(\rho^n_0,\rho^n_1)$ is a Cauchy sequence. For this, by the triangle inequality, we only need to show $W_{2,p}(\rho_i^n, \rho_i^m) \to 0$ as $n, m \to \infty$ for $i = 0,1$. For $\varepsilon \in (0,1)$, 
we define $\w{\rho}_0 = (1- \varepsilon) \rho_0 + \varepsilon {\bf 1}$, and the linear interpolation $\gamma_n(s):= (1 - s) \rho_0^n + s \w{\rho}_0$ which satisfies $\gamma_n(s) \ge s \varepsilon {\bf 1}$. Then it is easy to see 
\begin{align*} 
e^{\pm \omega_j/2p}\Gamma_\si^{-1/\h{p}}(\gamma_n(s)) \ge \inf_j \{e^{-|\omega_j|/2p}\} s \varepsilon \sigma^{-1/\h{p}}\,,
\end{align*}
by which, Lemma \ref{lem:mono_norm} implies that $[\gamma_n(s)]_{p,\omega}^{-1} \le C (s \varepsilon)^{p-2} {\rm id}_{\mc{H}}$ holds for some constant $C > 0$, and hence that
\begin{align*}
    \mf{D}_{p,\gamma_n(s)}^{-1} \le C (s \varepsilon)^{p-2} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} {\rm id}_{\mc{H}}\,.
\end{align*}
Recalling the expression \eqref{def:wp3}, we obtain 
\begin{align} \label{auxeq:bry}
    W_{2,p}(\rho_0^n, \w{\rho}_0) &\le \int_0^1 \left\langle \dot{\gamma}_n(s), \mf{D}_{p,\gamma_n(s)}^{-1} \dot{\gamma}_n(s)  \right\rangle^{1/2}\, ds \notag \\
    & \le C (s \varepsilon)^{p-2} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} \int_0^1 \norm{ \dot{\gamma}_n(s)}_2\, ds\,.
\end{align}
Note the estimate:
\begin{align*}
    \norm{ \dot{\gamma}_n(s)}_2 = \norm{\w{\rho}_0 - \rho^n_0}_2 \le \varepsilon \norm{\rho_0 - {\bf 1}}_2 + \norm{\rho_0 - \rho_0^n}_2\,,
\end{align*}
which, by \eqref{auxeq:bry}, gives
\begin{align*}
   \limsup_{n \to \infty}  W_{2,p}(\rho_0^n, \w{\rho}_0) \le C (s \varepsilon)^{p-1} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} \norm{\rho_0 - {\bf 1}}_2\,.
\end{align*}
Since $p \in (1,2]$ and $\varepsilon$ is arbitrary, the above estimate implies $ \lim_{n,m \to \infty} W_{2,p}(\rho_0^n, \rho_0^m) = 0$, by the triangle inequality.
\end{proof} 

\begin{proof}[Proof of Theorem \ref{thm:main_wasser}]
We will first show the existence of the minimizer of \eqref{def:wp4} by a direct method. Let $\left\{(\gamma^{(n)},{\bf B}^{(n)})\right\}$ be the minimizing sequence such that 
$
\sup_n \int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}}\ ds < + \infty\,.
$
We claim that there exists constant $C_j$, depending on $\sigma \in \mathcal{D}_+(\mc{H})$ and $\omega_j \in \mathbb{R}$, such that 
\begin{align} \label{claim_one}
\langle X, [\rho]_{p,\omega_j} X\rangle \le C_j \norm{X}_2^2\,,\quad \forall \rho \in \mathcal{D}(\mc{H})\,.
\end{align}
To show this, by the representation \eqref{eq:repre_rpw} of $[\rho]_{p,\omega}$ with notations from Proposition \ref{prop:repkernel}, we first have 
\begin{align*}
    \langle X,  [\rho]_{p,\omega} X \rangle \le (p-1)  \bigg( \sup_{i,k} \theta_p\left(e^{\omega/2p}\lambda_{k,p}, e^{-\omega/2p}\lambda_{i,p}\right)
   
    \bigg) \norm{\Gamma_\si^{1/\h{p}}(X)}_2^2\,.
\end{align*}
We also note that there exists a closed interval $I$ containing $e^{\pm \omega/2p} \lambda_{k,p}$ for all $\rho \in \mathcal{D}(\mc{H})$, and that the function $\theta_p(x,y)$ is bounded for $x,y \in I$. Then the claim \eqref{claim_one} readily follows. Since ${\bf B}^{(n)} \in L^1\left([0,1],\mathcal{B}(\mc{H})^J\right)$, we
can consider the $\mathcal{B}(\mc{H})^J$-valued measure $\mathsf{B}^{(n)}(ds) := {\bf B}^{(n)}(s) ds$. Then, for every Borel set $E \subset [0,1]$, we estimate 
\begin{align} \label{eq:estimate_minimizing}
    |\mathsf{B}^{(n)}|(E) & \le \int_E \bigg(\sum_{j = 1}^J \norm{B_j^{(n)}(s)}_2^2 \bigg)^{1/2} ds \notag \\
    & = \int_E \bigg(\sum_{j = 1}^J \left\langle [\gamma^{(n)}]_{p,\omega_j}^{-1/2} B_j^{(n)}, [\gamma^{(n)}]_{p,\omega_j} [\gamma^{(n)}]_{p,\omega_j}^{-1/2} B_j^{(n)} \right\rangle \bigg)^{1/2} ds \notag  \\
    & \le C \mathscr{L}(E)^{1/2} \bigg( \int_E \sum_{j = 1}^J \left\langle  B_j^{(n)}, [\gamma^{(n)}]^{-1}_{p,\omega_j}  B_j^{(n)} \right\rangle  ds \bigg)^{1/2}  \notag  \\
     & \le C \mathscr{L}(E)^{1/2} \bigg(\int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}} ds \bigg)^{1/2} < +\infty\,,
\end{align}
for $C := \max_j\{C_j^{1/2}\}$, where $\mathscr{L}$ is the Lebesgue measure on $\mathbb{R}$ and the third line is from H\"{o}lder's inequality and the estimate \eqref{claim_one}. 
It follows that the total variations of the measures $\mathsf{B}^{(n)}$ are uniformly bounded in $n$. Hence there exists a subsequence, still denoted by $\mathsf{B}^{(n)}$, converging weakly* to a $\mathcal{B}(\mc{H})^J$-valued measure $\mathsf{B}_*$. Then, by \eqref{eq:estimate_minimizing}, we can obtain
\begin{equation*}
    |\mathsf{B}_*|(E) \le \liminf_{n \to \infty}  |\mathsf{B}^{(n)}|(E) \le C \mathscr{L}(E)^{1/2}\,,
\end{equation*}
for some constant $C$, which implies that $\mathsf{B}_* \ll \mathscr{L}$ and hence the R-N derivative ${\bf B}_*: = \frac{d \mathsf{B}_*}{d \mathscr{L}}$ exists. We next prove that $\gamma^{(n)}(s)$ converges pointwise to some $\gamma(s): [0,1] \to \mathcal{D}(\mc{H})$. For this, we note 
\begin{equation*}
    \gamma^{(n)}(t) - \gamma^{(n)}(0) = - \int_0^t {\rm div} {\bf B}^{(n)}(s)\ ds. 
\end{equation*}
Then by the weak* convergence of $\mathsf{B}^{(n)}$ and $\gamma^{(n)}(0) = \rho_0$, we have the pointwise convergence of $\gamma^{(n)}(s)$ with the limit denoted by $\gamma_*(s)$.
Moreover, it is easy to check that $\gamma_*(s) \in C([0,1],\mathcal{D}(\mc{H})) $ and ${\bf B}_* \in L^1([0,1],\mathcal{B}(\mc{H})^J)$ satisfy the continuity equation. By dominated convergence theorem, we also have $\mc{B}(\mathcal{B}(\mc{H}))$-valued measure $[\gamma^{(n)}(s)]_{p,\omega_j} ds$ weakly* converge to $[\gamma_*(s)]_{p,\omega_j} ds$. Finally, noting the integral representation for $[\rho]^{-1}_{p,\omega}$ from \eqref{eq:inte_divi_p} and \eqref{def:kernel_multiplication}: 
\begin{align*}
      [\rho]^{-1}_{p,\omega}(\cdot)
   
   
   
    & =
    \frac{\sin((p-1) \pi)}{\pi} \Gamma_\si^{-1/\h{p}} \int_0^\infty s^{p - 2}   g_s^{[1]} \left(e^{\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho),  e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho) \right)  \Gamma_\si^{-1/\h{p}}(\cdot)\ ds\,,
   
   
\end{align*}
 we have 
\small
\begin{align} \label{auxeq_final}
  & \liminf_{n \to \infty} \int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}}\ ds  
  \notag \\
  \ge & \frac{\sin((p-1)\pi)}{\pi} \sum_{j=1}^J \int_0^\infty t^{p-2} \liminf_{n \to \infty} \int_0^1  \left\langle \Gamma_\si^{-1/\h{p}}\big(B_j^{(n)}(s)\big) , g_t^{[1]} \left(e^{\omega/2p}  \Gamma_\si^{-1/\h{p}}\big(\gamma^{(n)}(s)\big),  e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}\big(\gamma^{(n)}(s)\big) \right)  \Gamma_\si^{-1/\h{p}}\big(B_j^{(n)}(s)\big) \right\rangle ds dt    \notag \\
   \ge & \frac{\sin((p-1)\pi)}{\pi} \sum_{j=1}^J \int_0^\infty t^{p-2} \int_0^1  \left\langle \Gamma_\si^{-1/\h{p}}\big(B_{*,j}(s)\big) , g_t^{[1]} \left(e^{\omega/2p}  \Gamma_\si^{-1/\h{p}}\big(\gamma_*(s)\big),  e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}\big(\gamma_*(s)\big) \right)  \Gamma_\si^{-1/\h{p}}\big(B_{*,j}(s)\big) \right\rangle ds dt   \notag \\
  \ge  & \int_0^1 \big\lVert{\bf B}_*\big\rVert^2_{-1,p,\gamma_*} ds\,,
\end{align}
\normalsize
where in the first inequality we have used Fatou's lemma, and in the second inequality we have used Theorem 3.4.3 of \cite{buttazzo1989semicontinuity} on the lower-semicontinuity of integral functionals. The estimate \eqref{auxeq_final} directly implies that $(\gamma_*, {\bf B}_*)$ is a minimizer to \eqref{def:wp4}. 

Recalling formulations \eqref{def:wp1} and \eqref{def:wp3}, by Jensen's inequality, we find 
\begin{align*}
W_{2,p}(\rho_0,\rho_1) =  \Big(\int_0^1 g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right) ds \Big)^{1/2} = \int_0^1 g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right)^{1/2} ds\,.
\end{align*}
It follows that $W_{2,p}(\rho_0,\rho_1)^2 = g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right)$ for almost everywhere $s \in [0,1]$. Then, by definition of $W_{2,p}$ and a time scaling, it is easy to check that the property \eqref{eq:geo_mini} holds. Therefore, we have proved that $(\mc{D}(\mc{H}), W_{2,p})$ is a geodesic space. The completeness of the metric space $(\mc{D}(\mc{H}), W_{2,p})$ is a simple consequence of Lemma \ref{lem:lower_bound}. Indeed, let $\{\rho_n\}$ be a Cauchy sequence such that $W_{2,p}(\rho_n,\rho_m) \to 0$ as $n,m \to \infty$. Then, by Lemma \ref{lem:lower_bound}, it is also Cauchy in the complete metric space $(\mathcal{D}(\mc{H}),\norm{\cdot}_1)$, which implies that there exists $\rho_\infty \in \mathcal{D}(\mc{H})$ such that $\norm{\rho_n - \rho_\infty}_1 \to 0$ as $n \to \infty$. Similarly to the proof of Lemma \ref{lem:exten_bry}, by $\norm{\rho_n - \rho_\infty}_1 \to 0$, we can conclude $W_{2,p}(\rho_n,\rho_\infty) \to 0$ as $n \to \infty$. 
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:lower_bound}]
For any $\delta > 0$ and $\rho_0, \rho_1 \in \mathcal{D}_+(\mc{H})$, by Theorem \ref{thm:main_wasser}, there exists the curve $(\gamma(s),{\bf B}(s))$, $s \in [0,1]$, satisfying $\dot{\gamma}(s) + {\rm div} {\bf B}(s) = 0$ with  $\gamma(0) = \rho_0$ and $\gamma(1) = \rho_1$, such that
\begin{align*}
    \int_0^1 \norm{{\bf B}(s)}^2_{-1, p, \gamma(s)} ds \le W_{2,p}(\rho_0,\rho_1)^2 + \delta\,.
\end{align*}
It then follows that, by Cauchy's inequality,
\begin{align} \label{auxeq_bound1}
    \tr(X(\rho_1 - \rho_0))  = \tr \left(X \int_0^1 \dot{\gamma}(s) ds \right) \notag
     & = \int_0^1  \langle \nabla X, {\bf B}(s) \rangle ds \notag \\
    &\le \left(\int_0^1 \norm{\nabla X}_{p,\gamma(s)}^2 d s \right)^{1/2} \left( \int_0^1 \norm{{\bf B}(s) }_{-1,p,\gamma(s)}^2 d s \right)^{1/2} \notag \\
    &\le \left(\int_0^1  \sum_j \langle \partial_j X, [\gamma(s)]_{p,\omega_j} \partial_j X  \rangle  d s \right)^{1/2} \left( W_{2,p}(\rho_0,\rho_1)^2 + \delta \right)^{1/2}. 
\end{align}

To deal with the term $\langle \partial_j X, [\gamma(s)]_{p,\omega_j} \partial_j X  \rangle$, we next estimate the kernel operator $[\rho]_{p,\omega}$ for $\rho \in \mc{D}_+$ and $\omega \in \mathbb{R}$. For this, recalling \eqref{eq:repre_rpw}:
\begin{align} \label{eq:kernel2}
      [\rho]_{p,\omega}(\cdot) = (p-1)\sum_{i,k}\frac{e^{\omega/2p}\lambda_{k} - e^{-\omega/2p}\lambda_{i}}{\left(e^{\omega/2p}\lambda_{k}\right)^{p-1} - \left(e^{-\omega/2p}\lambda_{i}\right)^{p-1}} \Gamma_\si^{1/\h{p}} \left(E_{k} \Gamma_\si^{1/\h{p}}(\cdot) E_{i}\right),
\end{align}
where $\lambda_{i}$ and $E_{i}$ are eigenvalues and the associated eigen-projections of $\Gamma_\si^{-1/\h{p}}(\rho)$, respectively (we omit the subscript $p$ of $\lambda_i$ and $E_i$ for simplicity). 
By the integral presentation \eqref{eq:inte_divi_p}, we can estimate 
\begin{align} \label{eq:ker_estimate}
  &\frac{1}{p-1}  \frac{\left(e^{\omega/2p}\lambda_{k}\right)^{p-1} - \left( e^{-\omega/2p}\lambda_{i}\right)^{p-1}}{e^{\omega/2p}\lambda_{k} - e^{-\omega/2p}\lambda_{i}}
\notag \\  
= &  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} 
\frac{\log (s + e^{\omega/2p} \lambda_k) - \log (s + e^{-\omega/2p}\lambda_i)  }{e^{\omega/2p} \lambda_k - e^{-\omega/2p}\lambda_i}
\  ds \notag\\
\ge &  \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty  s^{p-2} \frac{2}{2s + e^{\omega/2p} \lambda_k + e^{-\omega/2p}\lambda_i}\  ds \notag \\
= & \frac{\sin((p-1) \pi)}{\pi} \left(e^{\omega/2p}\lambda_k + e^{-\omega/2p} \lambda_i\right)^{p-2} \int_0^\infty s^{p-2}  \frac{2}{1+ 2 s}\  ds\,, 
\end{align}
where the third line is from the elementary inequality: 
\begin{equation*}
    \frac{x - y}{\log x - \log y} \le \frac{x + y}{2}\quad \text{for all}\  x, y > 0\,,
\end{equation*}
and the last line is by the change of variable.  We define the constant:
\begin{equation}\label{def:cons_cp}
     C_p := \frac{\sin((p - 1) \pi)}{\pi}  \int_0^\infty s^{p-2}  \frac{2}{1+ 2 s} \ d t\,. 
\end{equation}
By \eqref{eq:kernel2} and \eqref{eq:ker_estimate}, there holds 
\begin{align*}
\left\langle \partial_j X,    [\rho]_{p,\omega}\partial_j X \right\rangle \le & C_p^{-1} \sum_{i,k} \left(e^{\omega/2p}\lambda_k + e^{-\omega/2p} \lambda_i\right)^{2-p}  \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X),  \left(E_{k} \Gamma_\si^{1/\h{p}}(\partial_j X) E_{i}\right) \right\rangle \\
    \le &  C_p^{-1} \sum_{i,k} \left(\left(e^{\omega/2p}\lambda_k \right)^{2-p} + \left(e^{-\omega/2p} \lambda_i\right)^{2-p}\right)   \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X),  \left(E_{k} \Gamma_\si^{1/\h{p}}(\partial_j X) E_{i}\right) \right\rangle \\ 
     = &  C_p^{-1}  \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X),  \left(L_{\left(e^{\omega/2p}\Gamma_\si^{-1/\h{p}}(\rho)\right)^{2-p}} + R_{\left(e^{-\omega/2p}  \Gamma_\si^{-1/\h{p}}(\rho) \right)^{2-p}}\right)  \Gamma_\si^{1/\h{p}}(\partial_j X)  \right\rangle \\
   
      \le &  C_p^{-1}  \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X),  \left(L_{\left(e^{\omega/2p}\sigma^{-1/\h{p}}\right)^{2-p}} + R_{\left(e^{-\omega/2p}  \sigma^{-1/\h{p}} \right)^{2-p}}\right)  \Gamma_\si^{1/\h{p}}(\partial_j X)  \right\rangle  \\
     \le & C_p^{-1} \left( e^{(2-p)\omega/2p} + e^{(p-2)\omega/2p}  \right) \tr \left( \sigma^{(p-2)/\h{p}} \right)  \norm{\Gamma_\si^{1/\h{p}}(\partial_j X)}_{\infty}^2 \\
     \le & 4 C_p^{-1} \left(e^{(2-p)\omega/2p} + e^{(p-2)\omega/2p}  \right) \tr \left( \sigma^{(p-2)/\h{p}} \right)  \norm{\sigma}_\infty^{2/\h{p}}\norm{V_j}^2_\infty \norm{X}^2_\infty\,,
\end{align*}
where the second inequality is by $(x + y)^p \le x^p + y^p$ for $p \in (0,1)$ and $x, y > 0$; the third inequality is by $\rho \le {\bf 1}$ for all $\rho \in \mathcal{D}(\mc{H})$ and the operator monotonicity of $t^p$ for $0 \le p \le 1$; the fourth inequality is by
H\"{o}lder's inequality.  Then we arrive at, by \eqref{auxeq_bound1}, 
\begin{align} \label{eq:est_lower}
     \norm{\rho_1 - \rho_0}_1 & \le \left( 4 C_p^{-1} \tr \left( \sigma^{(p-2)/\h{p}} \right) \norm{\sigma}_\infty^{2/\h{p}} \sum_j \left(e^{(2-p)\omega_j/2p} + e^{(p-2)\omega_j/2p}  \right) \norm{V_j}_\infty^2 \right)^{1/2}  W_{2,p}(\rho_0,\rho_1)\,. 
\end{align}
 
We finally prove the uniform boundedness of the prefactor in \eqref{eq:est_lower} for $p \in (1,2]$. It suffices to consider the constant $C_p$ \eqref{def:cons_cp}. By elementary calculation, we derive 
\begin{align*}
\frac{2}{3}\left( \frac{1}{p - 1} + \frac{1}{2 - p} \right) \le \int_0^\infty \frac{2 s^{p-2} }{1+ 2 s} d s  &= \int_0^1  \frac{2 s^{p-2}}{1+ 2 s} d s + \int_1^\infty   \frac{2 s^{p-2}}{1+ 2 s} d s \le \frac{2}{p - 1} + \frac{1}{2 - p}\,,
\end{align*}
which immediately gives the following estimates: 
\begin{align*}
\frac{2}{3} + O(p - 1) \le  C_p \le 2 + O(p-1) \quad \text{as} \ p \to 1^+\,,
\end{align*}
and 
\begin{align*}
\frac{2}{3} + O(p - 1) \le  C_p \le 1 + O(p-2) \quad \text{as} \ p \to 2\,.
\end{align*}
The proof is complete. 
\end{proof}



\end{appendix}

