\section{Introduction}

In this study, we consider the following $N$-particle mean-field Schr\"odinger equation
\begin{equation}\label{eq:Schrodinger_0}
\begin{split}
	\mathrm{i} \hbar \partial_t\, \psi_{N,t} &=-\frac{\hbar^2}{2} \sum_{j=1}^{N}  \Delta_{x_j} \psi_{N,t} + \frac{1}{2N} \sum_{i\neq j}^{N} V(x_i-x_j) \psi_{N,t}\\
	\psi_{N,0} &= \frac{1}{\sqrt{N!}} \det{\mathrm{e}_j (x_i)}_{i,j=1}^{N},
\end{split}
\end{equation}
where $\{\mathrm{e}_j\}_{j=1}^N$ is a family of orthonormal basis in $L^2(\mathbb{R}^3)$ and $\Delta_{x_j}$ is the Laplacian on $j$-th particle.
The initial data in \eqref{eq:Schrodinger_0} is in the form of a Slater determinant, which stays in the antisymmetric subspace $L^2_a(\bR^{3N})$ of $L^2(\mathbb{R}^3)$ with $\norm{\psi_{N,t}}_2=1$, where
\[
L^2_a(\bR^{3N}) := \big\{ \psi_N \in L^2(\bR^{3N}) :\ \psi_N(x_{\pi(1)},\dots,x_{\pi(N)}) = (\sign{\pi})  \psi_N(x_1, \dots, x_N)\ \text{for all } \pi \in S_N\big\}.
\]
In the above formulation, $S_N$ is the odd-permutation group.

The justification of such a choice is stemmed from the fact that, due to the Pauli's exclusion principle,
the kinetic energy for $N$-fermionic particles is of order $\hbar^2 N^{5/3}$.
Moreover, to make the potential energy to be comparable with the kinetic energy, the coupling constant in front of the interaction is taken to be of the order $\hbar^2 N^{-1/3}$. 
Then because of average velocity of particles is of order $\hbar N^{-1/3}$, we consider a short time scale, i.e.,
\[
\hbar = N^{-1/3}.
\]
We call such scaling as semi-classical scale.
For more details, we refer to \cite{BACH20161, benedikter2014mean, Chen2021AHP}.

As it is difficult to solve the Schr\"odinger equation in \eqref{eq:Schrodinger_0} numerically when the number of particle $N$ is large, we aim to derive its corresponding effective evolution equation.
In fact, we consider the $k$-particle reduced density matrix where its corresponding integral kernel is given by
\begin{equation}\label{eq:gammak}
	\begin{aligned}
		&\gamma_{N,t}^{(k)} (x_1,\dots,x_k;y_1,\dots,y_k)\\
		&= \frac{N!}{(N-k)!} \int \mathrm{d}{x_{k+1}} \dots \mathrm{d}{x_N}\, \overline{\psi_{N,t}(y_1,\dots,y_k,x_{k+1},\dots,x_N)} \psi_{N,t}(x_1, \dots, x_k,x_{k+1},\dots,x_N),
	\end{aligned}
\end{equation}
where $1\leq k \leq N$. Moreover, we denote the expectation of the one-particle observable as follows,
\[
	\left< \psi_{N,t}, O \psi_{N,t} \right> = \int \mathrm{d}{x_2} \dots \mathrm{d}{x_N}\, \overline{\psi_{N,t}(y_1,x_{2},\dots,x_N)} \big(O \psi_{N,t}\big)(x_1,x_{2},\dots,x_N),
\]
which can be also written as follows:
\[
	\Tr O\gamma^{(1)}_{N,t} = \left< \psi_{N,t}, O \psi_{N,t}\right>.
\]
The one-particle reduce density matrix of the initial data given in \eqref{eq:Schrodinger_0} is
\[
	\omega_N = \sum_{j=1}^N \dyad{\mathrm{e}_j}{\mathrm{e}_j},
\]
where its corresponding integral kernel is $\omega_N(x;y) = \sum_{j=1}^N \overline{\mathrm{e}_j(y)} \mathrm{e}_j(x)$.

\subsubsection{Short review of mean-field limit, $N\to \infty$}

It is well known that the Hartree-Fock equation 
\begin{equation}\label{eq:hartree-fock}
	\begin{split}
		\mathrm{i} \hbar \partial_t \, \omega_{N,t} &= \big[-\hbar^2 \Delta + (V*\rho_{N,t}) - X_t, \omega_{N,t}\big],\\
        \rho_{N,t} &= \frac{1}{N} \omega_{N,t}(x;x) \\
        X_t &= \frac{1}{N} V(x-y) \omega_{N,t}(x;y)\\
	\end{split}
\end{equation}
is used to approximate the Schr\"odinger equation in the mean-field limit.
Here we denote $[A,B]:=AB-BA$.

The convergence for fixed $\hbar$ has been given in \cite{ELGART20041241} for short time.
Under the scaling $\hbar = N^{-1/3}$, the rates of convergence in the trace norm and the Hilbert-Schmidt norm are obtained for arbitrary given time in \cite{benedikter2014mean} when the initial data is an approximation of the Slater determinant. Later on,  the case with mixed state initial data has been considered in \cite{benediktermixed,benedikter2014rel}. Furthermore, for Coulomb and Riesz potentials, the rate of convergence is obtained in \cite{Porta2017,saffirio2017mean}. We refer more references on this topic to \cite{Frohlich2011,petrat2014derivation,Petrat2017,Petrat2016ANM} and the references therein.

\subsubsection{Short review of semi-classical limit, $\hbar\to0$}

The Vlasov equation can be obtained via semi-classical limit and from the Hartree or the Hartree--Fock equations. It has been first investigated in \cite{Lions1993} by using Wigner measure.
Recently, the rates of convergence in the trace norm as well as the Hilbert-Schmidt norm has been studied in \cite{benedikter2016hartree} with regular assumptions on the initial data.
The $k$-particle Wigner measure
\begin{equation}\label{eq:wigner_measure}
	\begin{aligned}
		&W^{(k)}_{N,t}(x_1,p_1, \dots, x_k, p_k) \\
		&:= \binom{N}{k}^{-1} \int (\dd{y})^{\otimes k} \; 
		\gamma_{N,t}^{(k)}\left(x_1 + \frac{\hbar}{2}y_1, \dots, x_k + \frac{\hbar}{2}y_k; x_1 - \frac{\hbar}{2}y_1, \dots, x_k - \frac{\hbar}{2}y_k \right)e^{-\mathrm{i} \sum_{i=1}^k p_i \cdot y_i},
	\end{aligned}
\end{equation}
has been used to study semi-classical limit, (see e.g. in \cite{benedikter2016hartree, saffirio2019hartree}).
Here $\gamma_{N,t}^{(k)}$ is the kernel of the $k$-particle reduced density matrix defined in \eqref{eq:gammak}.

Some of the recent developments in the semi-classical limit are the following: One can find results 
for the inverse power law potential in \cite{Saffirio2019}, for the rate of convergence in the Schatten norm in \cite{lafleche2020strong}, for the Coulomb potential and mixed states in \cite{saffirio2019hartree}, and for the convergence in the Wasserstein distance in \cite{Lafleche2019GlobalSL,Lafleche2019PropagationOM}.
Relativistic fermionic system has been studied in \cite{Dietler2018}. Further analyses of the semi-classical limit can be found in \cite{amour2013classical,amour2013,Athanassoulis2011,Gasser1998, Markowich1993}.

\subsubsection{Combined Mean-Field and semi-classical Limits}
Narnhofer and Sewell, and Spohn independently derived  Vlasov equation~\eqref{eq:vlasov} from the $N$-body Schrödinger equation \eqref{eq:Schrodinger_0} with $\hbar = N^{-1/3}$, in \cite{Narnhofer1981,Spohn1981}.
Without assuming $\hbar = N^{-1/3}$, a rate of convergence was obtained in \cite{graffi2003mean} in a weak formulation.
The rate of convergence of the combined limits was studied in \cite{Golse2017,Golse2021,golse:hal-01334365} by using the Wasserstein (pseudo-)distance.
Under a generalized Husimi measure framework, the authors in \cite{Chen2021JSP} obtained the convergence for regular potentials, and then they also considered a cut-offed potential which converges to Coulomb type singularity in the large $N$ limit in \cite{Chen2021AHP}.
Recently, the combined limit for the singular potential case was obtained in \cite{Chong2021}.

It is known that the Wigner measure in \eqref{eq:wigner_measure} is not a (proper) probability density, as there might be some point having negative sign. (We refer, e.g.,  ~\cite{doi:10.1063/1.531326,CaseWignerpedestrians,Hudson1974,kenfack2004negativity,PhysRevA.79.062302,doi:10.1063/1.525607} for further references on Wigner measure.)
It has been shown that the Husimi measure, the convolution of the Wigner measure with a Gaussian function, is a nonnegative probability measure
\cite{Combescure2012,Fournais2018,Zhang2008}.
In particular, from \cite[p.21]{Fournais2018}, given a specific Gaussian coherent state, the relation between the Husimi measure and Wigner measure is given by the following convolution: for any $1 \leq k \leq N$,
\begin{equation}\label{eq:gaussian_m_vs_W}
	\mathfrak{m}_{N,t}^{(k)}  := \frac{N(N-1)\cdots (N-k+1)}{N^k} W^{(k)}_{N,t} * \mathcal{G}^\hbar,
\end{equation}
where
\begin{equation*}
	\mathcal{G}^\hbar (q_1,p_1,\dots, q_k,p_k) := \frac{1}{(\pi \hbar)^{3k} }\exp \Big(-\frac{ \sum_{j=1}^k {q_j}^2+ {p_j}^2}{\hbar} \Big).
\end{equation*}

In this study, we will consider the following generalized  $k$-particle Husimi measure:
For any $p,q \in \bR^3$ and $\psi_{N,t} \in L^2_a(\bR^{3N})$, the $k$-particle Husimi measure is given by
\begin{equation}\label{eq:husimi_def_1}
	m_{N,t}^{(k)} (q_1,p_1,\dots, q_k, p_k) = \langle \psi_{N,t}, a^*(f^\hbar_{q_1, p_1}) \cdots a^*(f^\hbar_{q_k, p_k}) a(f^\hbar_{q_k, p_k}) \cdots  a(f^\hbar_{q_1, p_1}) \psi_{N,t} \rangle.
\end{equation}
Here $a^*(f^\hbar_{q,p})$ and $a(f^\hbar_{q,p})$ are standard creation- and annihilation-operator respectively\footnote{Definitions of creation- and annihilation-operator is provided in Appendix \ref{sec:FockSpace}.} with respect to the coherent state $f^\hbar_{q,p}$ given by
\begin{equation*}\label{eq:def_coherent}
	f^{\hbar}_{q, p} (y) := \hbar^{-\frac{3}{4}} f \left(\frac{y-q}{\sqrt{\hbar}} \right) e^{\frac{\mathrm{i}}{\hbar} p \cdot y },
\end{equation*}
where $f$ is any given real-valued function satisfying $\norm{f}_2 =1$.


\begin{Remark}
	As stated in \cite{Fournais2018}, the $k$-particle Husimi measure here describes how many fermions are within the $k$-semi-classical boxes of length $\sqrt{\hbar}$ centered at the phase-spaces $(q_1,p_1), \dots, (q_k,p_k)$.
\end{Remark}

\begin{Remark}
	If $f (x)= \pi^{-3/4} e^{-|x|^2/2}$, \cite{Combescure2012} shows that the $k$-particle Husimi measure $m_{N,t}^{(k)}$ coincides with the $\mathfrak{m}^{(k)}_{N,t}$ in \eqref{eq:gaussian_m_vs_W}.
\end{Remark}

\subsection{Main Result}
Denoting the one-particle Husimi measure as $m_{N,t} := m_{N,t}^{(1)}$ and assume that $\psi_{N,t}$ is the solution to the Schr\"odinger equation in \eqref{eq:Schrodinger_0}, from  \cite[Proposition 2.1]{Chen2021JSP}, we obtain the following identity:
\begin{equation} \label{eq:Chen2021JSP_representation}
	\begin{aligned}
		\partial_t m_{N,t}(q,p) &+ p \cdot \nabla_{q} m_{N,t}(q,p) - \nabla_q\cdot\left(\hbar \Im \langle \nabla_{q} a (f^\hbar_{q,p}) \psi_{N,t}, a (f^\hbar_{q,p}) \psi_{N,t} \rangle\right)\\
		&= \frac{1}{(2\pi)^3}\nabla_p\cdot\int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2 \mathrm{d}q_2 \mathrm{d}p_2\,    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2}\\
		& \hspace{1.5cm}\int_0^1 \mathrm{d}s\ \nabla V_N\big( su_1+ (1-s)w_1-w_2 \big)  \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2),
	\end{aligned}
\end{equation}
where
\[
    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2} :=  f^\hbar_{q,p}(w_1)  \overline{f^\hbar_{q,p}(u_1)}  f^\hbar_{q_2,p_2}(w_2)  \overline{f^\hbar_{q_2,p_2}(u_2)}.
\]
	
\begin{Remark}
	The two-particle reduced density matrix in \eqref{eq:Chen2021JSP_representation} stems from \eqref{eq:k_reduced}.
\end{Remark}
	
Our aim is therefore to obtain the convergence from $m_{N,t}$, in weak sense, to the solution of Vlasov equation $m_t$ as follows:
\begin{equation}\label{eq:vlasov}
	\begin{split}
			\partial_t m_{t}(q,p) &= -p \cdot \nabla_q m_t(q,p) + \nabla_q \big(V * \varrho_{t}\big)(q) \cdot \nabla_p  m_t(q,p),\\
			m_{t}(q,p)\big|_{t=0} &= m_0 (q,p),
	\end{split}
\end{equation}
where the initial data $m_0 (q,p)$ is the one-particle Husimi measure with Slater determinant as the wave-function and 
\[
\varrho_{t} = \int \mathrm{d}p\,  m_t(q,p).
\]

We give the following assumptions in this paper.
\begin{assump}\label{assumptions}\phantom{text}
\begin{enumerate}
	\item (Interaction potential) Assume the interaction potential $V\in L^1(\bR^3)$ be a real-valued function such that $V(-x) = V(x)$ and $\int \mathrm{d}{p} (1+|p|^2) |\widehat{V}(p)| < \infty$, where $\widehat{V}$ is its Fourier transform of $V$.
		
	\item (Coherent sate) The real-valued function $f \in H^1(\bR^3)\cap L^\infty(\bR^3)$ satisfies $\norm{f}_2 = 1$, and has compact support in $B_{R_1}$.
		
	\item (Initial data) Let $m_{N}$ be the initial one-particle Husimi measure of the given initial data in \eqref{eq:Schrodinger_0} which converges to $m_0$ weakly in  $L^1$. Furthermore, it satisfies
	\begin{equation}\label{eq:inip2q_ben}
		\int \mathrm{d}q \mathrm{d}p\, (|p|^2 + |q|) m_{N}(q,p) <\infty.
	\end{equation}

	\item (Initial data) Let $\omega_N$ be the one-particle density matrix of initial data which satisfies
	\begin{equation}\label{eq:assum_semi-classical}
		\begin{aligned}
			\sup_{p\in\mathbb{R}^{3}}\frac{1}{1+|p|}\trnorm{[e^{\mathrm{i} p\cdot x}, \omega_{N}]} &\leq CN\hbar,\\
			\trnorm{[\hbar \nabla, \omega_{N}]} &\leq CN\hbar,
		\end{aligned}
	\end{equation}
	where $\trnorm{\cdot}$ is the trace norm.
\end{enumerate}
\end{assump}

\begin{Remark}
The assumptions in \eqref{eq:assum_semi-classical} can be explained by the nature of the semi-classical structure. More details can be found in \cite{benedikter2014mean} where mean field limit has been studied.
\end{Remark}

With the assumptions presented above, we have the following theorem:

\begin{Theorem} \label{thm:main_ben}	
Let $m_{N,t}$ be the one-particle Husimi measure defined in \eqref{eq:husimi_def_1} with $\psi_{N,t}$ the solution of Schr\"odinger equation \eqref{eq:Schrodinger_0}, and suppose the aforementioned assumptions hold and $m_{t}$ is the solution of Vlasov equation in \eqref{eq:vlasov}. Then, for any given $T>0$, $(m_{N,t})_{N\in\mathbb{N}}$ converges to $m_t$ weakly (*) in $L^p ((0,T)\times \bR^3 \times \bR^3)$ for arbitrary 
$1\leq p \leq \infty$.
\end{Theorem}

Instead of using BBGKY hierarchy method in \cite{Chen2021JSP}, we are working only on the equation for one-particle reduced density matrix. The two-particle reduced density matrix term appeared in the aforementioned equation is treated directly by using the mean-field structure via Bogoliubov transformation method.\footnote{The literature on Bogoliubov transformation can be found in \cite{benedikter2014mean,Benedikter2016book,nam_qm2, solo_lect}.} Furthermore, as shown later in the proof,
the order (depending on $N$) of  the residual terms for the mean-field term is smaller than the one presented in \cite{Chen2021JSP}.
	
Lastly, by refraining from using the hierarchy method, we can shed more light on the structure of the residual terms for both mean-field limit and semi-classical limit.
	
\vspace{1em}
\noindent \textbf{Idea of the proof:}

For convenience, we use the Fock space formalism, which will be briefly introduced in Appendix \ref{sec:FockSpace}.
By using Husimi transform given in \eqref{eq:Chen2021JSP_representation},
we transform the Schr\"odinger equation for $\psi_{N,t}$ into the Vlasov type equation for $m_{N,t}$  with residual terms. To be more precise, from the computations in \cite{Chen2021JSP}, we have
\begin{equation} \label{eq:BBGKY_k1_ben}
	\begin{aligned}
		&\partial_t m_{N,t}(q,p) + p \cdot \nabla_{q} m_{N,t}(q,p)\\
		&= \frac{1}{(2\pi)^3}\nabla_{p} \cdot  \int \dd{q_2} \nabla V(q- q_2) \varrho_{N,t}(q_2) m_{N,t}(q,p)
		+ \nabla_{q}\cdot \widetilde{\mathcal{R}} +\nabla_{p}\cdot \mathcal{R},
	\end{aligned}
\end{equation}
where $\varrho_{N,t}(q):= \int \dd{p} m_{N,t}(q,p)$, $\widetilde{\mathcal{R}}$ and $\mathcal{R}=\mathcal{R}_\mathrm{s}+\mathcal{R}_\mathrm{m}$ are given by
\begin{equation}\label{eq:bbgky_remainder_1_ben}
	\begin{aligned}
		\tilde{\mathcal{R}}_{\phantom{1}} &:=   \hbar \Im \left< \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t}, a (f^\hbar_{q,p}) \Psi_{N,t} \right>,
		\\
		{\mathcal{R}_\mathrm{s}} &:= \frac{1}{(2\pi)^3}  \int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2 \mathrm{d}q_2 \mathrm{d}p_2\,    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2}\\
		&
		\qquad \qquad  \bigg[\int_0^1 \mathrm{d}s\ \nabla V\big( su_1+ (1-s)w_1-w_2 \big)   - \nabla V(q-q_2) \bigg]\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  ,
		\\
		{\mathcal{R}_\mathrm{m}} &:= 	\frac{1}{(2\pi)^3}  \int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2 \mathrm{d}q_2 \mathrm{d}p_2\,    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2}\\
		&
		\qquad  \nabla V(q-q_2) \bigg[ \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2) -\gamma_{N,t}^{(1)}(u_1;w_1)\gamma_{N,t}^{(1)}(u_2;w_2)\bigg].
	\end{aligned}
\end{equation}

{Under the assumptions of Theorem \ref{thm:main_ben},
the estimate of the residual term $\tilde{\mathcal{R}}$ can be obtained exactly the same as in \cite{Chen2021JSP}.
(In the updated arXiv version of \cite{Chen2021JSP}, 
the oscillation estimates have been corrected, with which
the estimates for the residue terms from BBGKY hierarchy as well as the main result still hold true.)
In this paper we give a better estimate for the term $\tilde{\mathcal{R}}$, namely $\|{\int \mathrm{d}p\, \tilde{\mathcal{R}}}\|_{L^{\frac{5}{4}}(\mathbb{R}^3)}\leq C\hbar$. This result will be listed in Section \ref{sec:proof_mainTheorem}.}
 
Furthermore, as shown later  \ref{sec:mean_field_residue}, the estimate for semi-classical residue $\mathcal{R}_\mathrm{s}$ can be improved compared to \cite{Chen2021JSP}, by utilizing the finite moment estimate for kinetic energy.

For the mean-field residue term, we insert the intermediate terms in $\mathcal{R}_\mathrm{m}$ as follows:
\begin{equation}\label{eq:intro_insert}
	\begin{aligned}
		&\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  - \gamma_{N,t}^{(1)}(u_1;w_1)\gamma_{N,t}^{(1)}(u_2;w_2) \\
		&
		=  \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2) - \omega_{N,t}(u_1;w_1) \omega_{N,t}(u_2;w_2)\\
		&
		\quad + \big[ \omega_{N,t}(u_1;w_1)  -  \gamma^{(1)}_{N,t}(u_1;w_1)\big] \omega_{N,t}(u_2;w_2)\\
		&
		\quad + \gamma^{(1)}_{N,t}(u_1;w_1) \big[\omega_{N,t}(u_2;w_2) -  \gamma_{N,t}^{(1)}(u_2;w_2)\big]\\
		&
		=: T_1 + T_2 +T_3,
	\end{aligned}
\end{equation}
where $\omega_{N,t}$ is the solution to Hartree-Fock equation.

We observe that $T_2$ and $T_3$ are estimated by the trace norm and Hilbert-Schmidt norm of $\gamma_{N,t}^{(1)} - \omega_{N,t}$, respectively. As shown in Lemma \ref{lem:est_mean-field_osc}, the estimate for residue term involving $T_1$ will be controlled by using the fast oscillation effect from the coherent state. In this paper, we provide a bound of
\begin{equation}\label{eq:mixnorm}
	\left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \int \mathrm{d}{w_2} \left|\gamma_{N,t}^{(2)}(w_1, w_2; u_1, w_2)  - \omega_{N,t}(w_1;u_1) \omega_{N,t}(w_2;w_2)  \right| \right]^2   \right)^\frac{1}{2}.
\end{equation}

In \cite{benedikter2014mean}, the convergence with respect to the trace norm and Hilbert-Schmidt norm of the difference between $\gamma_{N,t}^{(k)}$ and $\omega_{N,t}^{(k)}$ are obtained separately with the help of Wick's theorem for $k \geq 2$. However, we do not directly use Wick's theorem to compute \eqref{eq:mixnorm} as extra effort is needed to estimate the residue term involving $T_1$. This is a different approach for the 2-particle reduced density matrix given in \cite{benedikter2014mean}. In particular, we trace the strategies given in \cite{benedikter2014mean} to obtain the rate of convergence estimate in the mean-field limit.
Then, the estimates are reduced to the expectation of the number operator $\mathcal{N}$ along the quantum fluctuation which will be bounded under the assumption of Theorem \ref{thm:main_ben}.

Furthermore, after concrete estimates for the residue terms, it will be proved that in the sense of distribution $\mathcal{R}_\mathrm{s}\sim \hbar^{\frac{1}{2}-}$ and $\mathcal{R}_\mathrm{m}\sim \hbar^{3-\frac{1}{2}-}$, from which one can observe that the semi-classical and mean field residue terms are not of the same order in the combined limit $N^{-1}=\hbar^3$ argument. The rate $\hbar^{\frac{1}{2}-}$ for semi-classical residue term should be of the optimal rate because of the semi-classical scale. The rate for mean-field residue we obtained here is not optimal. Nevertheless, to balance the two orders, one could think of other combined limit, which will be investigated in future projects.
		
This paper is arranged as follows: we first prove the estimate for semi-classical residue in Section \ref{sec:semi-classical_R}, followed by the estimate for mean-field residue in Section \ref{sec:mean_field_residue}. Finally, we will conclude the proof of Theorem \ref{thm:main_ben} in Section \ref{sec:proof_mainTheorem}.
In the appendices, for reader's convenience, we list some basic notations and known estimates.

{
\vspace{1.5em}	\noindent\textit{Acknowledgment}
We acknowledge support from the Deutsche Forschungsgemeinschaft through grant CH 955/4-1. 
Yue Li supported by National Natural Science Foundation of China (12071212).
Jinyeop Lee received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2111-390814868).	
}

\section{Estimate for mean-field residue}\label{sec:semi-classical_R}

In this section, we will estimate the mean-field residue by first showing in Lemma \ref{lem:est_mean-field_osc} that the estimate for mean-field residue term can be reduced to the estimate for the term
\begin{equation}\label{pdbmea}
    \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1)
\end{equation}
where we denote
\[
\Tr^{(1)} |\gamma^{(2)}  - \gamma^{(1)} \otimes \gamma^{(1)} | := \int \mathrm{d}{y} |\gamma^{(2)} (x,y;z,y) - \gamma^{(1)}(x;y) \gamma^{(1)}(z;y)|,
\]

Then, we prove the estimate for \eqref{pdbmea} in Proposition \ref{prop:mixed_norm} and finally summary the estimation for the mean-field residue in Corollary \ref{cor:est_mean-field_osc}.

\begin{Lemma}\label{lem:est_mean-field_osc}
Let $\varphi, \phi \in C_0^\infty(\bR^3)$. Then, for $\frac{1}{2} < \alpha_1 < 1$ and $s ={\left\lceil \frac{3(\alpha_1 \jinyeop{-} \frac{1}{2})}{2(1-\alpha_1)} \right\rceil}$, we have
\begin{equation}
	\begin{aligned}
		&\bigg|\int \mathrm{d}q \mathrm{d}p\, \varphi(q)\phi(p)  \nabla_{p} \cdot \mathcal{R}_\mathrm{m}(q,p) \bigg| \\
		&\quad \leq {C} \norm{\nabla V}_{L^\infty}  \hbar^{3+ \frac{3}{2}(\alpha_1 \jinyeop{-} \frac{1}{2}) + \frac{3}{2}}
		 \left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2   \right)^\frac{1}{2},
	\end{aligned}
\end{equation}
where the constant ${C}$ depends on $\norm{\varphi}_\infty$, $\norm{\nabla \phi}_{W^{s,\infty}}$, $\supp\phi$, $\norm{f}_{L^\infty \cap H^1}$, $\supp f$.
\end{Lemma}
\begin{proof}
Recall that, in \eqref{eq:bbgky_remainder_1_ben}, we defined the mean-field residue such that
\begin{equation}
	\begin{aligned}
		{\mathcal{R}_\mathrm{m}} &:= 	\frac{1}{(2\pi)^3}  \int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2 \mathrm{d}q_2 \mathrm{d}p_2\,    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2}\\
		&
		\qquad  \nabla V(q-q_2) \bigg[ \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2) -\gamma_{N,t}^{(1)}(u_1;w_1)\gamma_{N,t}^{(1)}(u_2;w_2)\bigg].
	\end{aligned}
\end{equation}
Then one obtains
\begin{align*}
	&\bigg|\int \mathrm{d}q \mathrm{d}p\, \varphi(q) \phi(p) \nabla_{p} \cdot \mathcal{R}_\mathrm{m} (q,p)\bigg| \\
	&
	= \frac{1}{(2\pi)^3}\bigg| \int (\mathrm{d}q \mathrm{d}p)^{\otimes 2} (\mathrm{d}w \mathrm{d}u)^{\otimes 2}\, \varphi(q) \nabla\phi(p)  \cdot  \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2} \nabla V(q-q_2) \\
	&
	\qquad \left[\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(u_2; w_2)\right] \bigg|\\
	&
	= \frac{1}{(2\pi\hbar)^3}\bigg| \int (\mathrm{d}q \mathrm{d}p)^{\otimes 2} (\mathrm{d}w \mathrm{d}u)^{\otimes 2}\,  \varphi(q) \nabla\phi(p)  \cdot \left(f\left(\frac{w-q}{\sqrt{\hbar}}\right) f\left(\frac{u-q}{\sqrt{\hbar}}\right) e^{\frac{\mathrm{i}}{\hbar} p\cdot(w-u)}\right)^{\otimes 2}  \nabla V(q-q_2) \\
	&
	\qquad \left[\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(u_2; w_2)\right] \bigg|\\
	&
	= \frac{1}{(2\pi\hbar)^3}\bigg| \int (\mathrm{d}q \mathrm{d}w \mathrm{d}u)^{\otimes 2}\, \left(f\left(\frac{w-q}{\sqrt{\hbar}}\right) f\left(\frac{u-q}{\sqrt{\hbar}}\right) \right)^{\otimes 2}\left(\int \mathrm{d}p\, \varphi(q) \nabla\phi(p)  \cdot e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}\right) \\
	&
	\qquad \nabla V(q-q_2) \left(\int \mathrm{d}p_2\, e^{\frac{\mathrm{i}}{\hbar} p_2\cdot(w_2-u_2)} \right) \left[\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  -  \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(u_2; w_2) \right] \bigg|\\
	&
	= \bigg| \int (\mathrm{d}q)^{\otimes 2} \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2 \,   f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right) \left|f\left(\frac{w_2-q_2}{\sqrt{\hbar}}\right)\right|^2  \left(\int \mathrm{d}p\, \varphi(q) \nabla\phi(p)  \cdot e^{\frac{\mathrm{i}}{\hbar} p_1\cdot(w_1-u_1)}\right) \\
	&
	\qquad \nabla V(q-q_2) \left[\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  -  \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right] \bigg|,
\end{align*}
where we use the weighted Dirac-Delta function in the last equality, i.e.,
\begin{equation}\label{eq:diract_delta}
	\frac{1}{(2\pi \hbar)^3}\int \mathrm{d}p_2\ e^{\frac{\mathrm{i}}{\hbar} p_2\cdot(w_2-u_2)} = \delta_{w_2}(u_2).
\end{equation}
Now, splitting the domains of $w_1$ and $u_1$ into two, namely, with the characteristic functions $\rchi_{(w_1 - u_1)\in \Omega_\hbar^\alpha}$ and $\rchi_{(w_1 - u_1)\in(\Omega_\hbar^\alpha)^c}$ whose domain $\Omega_\hbar^\alpha$ is defined in \eqref{eq:estimate_oscillation_omega}, we have
	
\begin{align*}
	&\leqslant \bigg| \int (\mathrm{d}q)^{\otimes 2}  \varphi(q) \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\   f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right) \left|f\left(\frac{w_2-q_2}{\sqrt{\hbar}}\right)\right|^2  \\
	&
	\qquad \left(\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in \Omega_\hbar^\alpha} e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)} \nabla\phi(p)   \right) \cdot \nabla V(q-q_2) \left[\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right] \bigg|\\
	&
	+  \bigg| 	\int (\mathrm{d}q)^{\otimes 2}\varphi(q) \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\    f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right) \left|f\left(\frac{w_2-q_2}{\sqrt{\hbar}}\right)\right|^2  \\
	&
	\qquad \left(\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c}  e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)} \nabla \phi(p)  \right) \cdot  \nabla V(q-q_2) \left[\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right] \bigg|\\
	& =: \text{I}_{\mathrm{m}} + \text{J}_{\mathrm{m}}.
\end{align*}
First, considering the term $\text{J}_{\mathrm{m}}$, we have
\begin{align*}
	\text{J}_{\mathrm{m}} & =  \bigg| 	\int \mathrm{d}q \mathrm{d}q_2\ \varphi(q) \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\   f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right)  \bigg|f\left(\frac{w_2-q_2}{\sqrt{\hbar}}\right)\bigg|^2 \\
	&
	\qquad \left(\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c} \nabla\phi(p) e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}\right) \cdot \nabla V(q-q_2) \left(\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right)\bigg|.
	\end{align*}
By the change of variable $\sqrt{\hbar}\,\tilde{q}_2 = w_2 - q_2$, we obtain
\begin{align*}
	& =  \bigg|	\int \mathrm{d}q\, \varphi(q) \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\, f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f \left(\frac{u_1-q}{\sqrt{\hbar}}  \right) \left(\hbar^{\frac{3}{2}} \int  \mathrm{d}\tilde{q}_2\   |f(\tilde{q}_2)|^2  \nabla V(q-w_2 + \sqrt{\hbar}\,\tilde{q}_2)\right) \\
	&
	\qquad \left(\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c} \nabla  \phi(p) e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}  \right)\left(\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right) \bigg|\\
	&
	\leq C \norm{\nabla V}_{L^\infty}  \hbar^{\frac{3}{2}} 	\int \mathrm{d}q\,  |\varphi(q)| \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\  \left|f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right) \right|\\
	&
	\qquad \left|\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c} \nabla  \phi(p) e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}\right| \left|  \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2) -  \gamma_{N,t}(u_1; w_1) \gamma_{N,t}(w_2; w_2) \right|\\
	&
	\leq C \norm{\nabla V}_{L^\infty}  \hbar^{\frac{3}{2}}   \int \mathrm{d}q\, | \varphi(q)| \int \mathrm{d}w_1\mathrm{d}u_1\,  \bigg| f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right)\bigg|\rchi_{|w_1-u_1|\leq 2R_1 \sqrt{\hbar}}\\
	&
	\qquad  \int\mathrm{d}w_2 \left|\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right|   \left|\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c} \nabla  \phi(p) e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}\right|.
\end{align*}
where we have used that supp$f\subset B_{R_1}$.
Recall again from Lemma \ref{lem:estimate_oscillation} that we have
\begin{equation}
	\left|\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in (\Omega_\hbar^\alpha)^c} e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)} \nabla  \phi(p) \right| \leq \norm{\nabla \phi}_{W^{s,\infty}}  \hbar^{(1-\alpha_1)s},
\end{equation}
for $s$ to be chosen later.
Then, continuing from $\text{J}_{\mathrm{m}}$,
\begin{align*}
	\text{J}_{\mathrm{m}} & \leq C \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty}  \hbar^{\frac{3}{2}+ (1-\alpha_1)s}  \int \mathrm{d}q\, |\varphi(q)|\int \mathrm{d}w_1\mathrm{d}u_1\,  \bigg|  f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right)\bigg| \\
	&
	\qquad  \Tr^{(1)} \left|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \right|(u_1;w_1)   \rchi_{|w_1-q|\leq R_1\sqrt{\hbar}}\;\rchi_{|u_1-q|\leq R_1 \sqrt{\hbar}},
\end{align*}
The H\"older's inequality yields
\begin{align*}
	\text{J}_{\mathrm{m}} & \leq C  \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty} \hbar^{\frac{3}{2}+ (1-\alpha_1)s} \int \mathrm{d}q\, |\varphi(q)| \left(\int \mathrm{d}w_1\mathrm{d}u_1\, \rchi_{|w_1-u_1|\leq 2R_1 \sqrt{\hbar}} \left|f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right)\right|^2 \right)^\frac{1}{2}\\
	&
	\qquad \left(\int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  -  \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big|(u_1;w_1) \right]^2 \rchi_{|w_1-q|\leq R_1\sqrt{\hbar}}\right)^\frac{1}{2}\\
	&
	\leq  C \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty} \hbar^{\frac{3}{2}+ (1-\alpha_1)s}
	\left(\hbar^3 \int \mathrm{d}\tilde{w}_1\mathrm{d}\tilde{u}_1\, \rchi_{|\tilde{w}_1-\tilde{u}_1|\leq 2R_1 } |f\left(\tilde{w}\right) f\left(\tilde{u}\right)|^2 \right)^\frac{1}{2}\\
	&
	\qquad \left(\int \mathrm{d}q  \int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  -  \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big|(u_1;w_1) \right]^2  \rchi_{|w_1-q|\leq R_1 \sqrt{\hbar}} \right)^\frac{1}{2}\\
	&
	\leq C   \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty}  \hbar^{ 3 + (1-\alpha_1)s } \left(\int \mathrm{d}\tilde{w}_1\mathrm{d}\tilde{u}_1\, |f\left(\tilde{w}\right) f\left(\tilde{u}\right)|^2\right)^\frac{1}{2} \\
	&
	\qquad \hbar^{\frac{3}{2}} \int \mathrm{d}\tilde{q}_1 \rchi_{|\tilde{q}_1|\leq R_1 }   \left(\int \mathrm{d}w_1\mathrm{d}u_1\,\left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big|(u_1;w_1) \right]^2  \right)^\frac{1}{2}\\
	&
	\leq C  \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty} \hbar^{ 3 + (1-\alpha_1)s+\frac{3}{2}} \left(\int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big |\gamma_{N,t}^{(2)}  -  \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big|(u_1;w_1) \right]^2  \right)^\frac{1}{2},
\end{align*}
where we denote
\[
	\Tr^{(1)} \left|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \right|(u_1;w_1) = \int \mathrm{d}{w_2} |\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2) - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)|.
\]
Thus, we have
\begin{equation}
	\text{J}_{\mathrm{m}}  \leq  C  \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty} \hbar^{ 3 + (1-\alpha_1)s+\frac{3}{2}}  \left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  -  \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big|(u_1;w_1) \right]^2  \right)^\frac{1}{2}.
\end{equation}
	
Now, we focus on $\text{I}_{m}$
\begin{align*}
	\text{I}_{\mathrm{m}}& =  (2 \pi )^3 \bigg| \int (\mathrm{d}q)^{\otimes 2} \varphi(q) \int \mathrm{d}w_1\mathrm{d}u_1\mathrm{d}w_2\,   f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right) \left|f\left(\frac{w_2-q_2}{\sqrt{\hbar}}\right)\right|^2  \\
	&
	\qquad \left(\int \mathrm{d}p\, \rchi_{(w_1 - u_1)\in \Omega_\hbar^\alpha}  e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)}  \nabla \phi(p) \right) \cdot \nabla V(q-q_2)\\
	&
	\qquad  \left[\gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)  - \gamma^{(1)}_{N,t}(u_1; w_1) \gamma^{(1)}_{N,t}(w_2; w_2)\right] \bigg|.
\end{align*}
Using the fact that
\[
	\left|\int \mathrm{d}p\,  e^{\frac{\mathrm{i}}{\hbar} p\cdot(w_1-u_1)} \nabla  \phi(p)\right|\leq \norm{\nabla \phi}_{L^1},
\]
we obtain the following estimate:
\begin{align*}
	\text{I}_{\mathrm{m}}& \leq C \norm{\nabla \phi}_{L^1}\norm{\nabla V}_{L^\infty} \int \mathrm{d}q\, |\varphi(q)|   \left( \int \mathrm{d}w_1\mathrm{d}u_1\, \rchi_{|w_1-u_1|\leq \hbar^{\alpha_1}}  \rchi_{|w_1-u_1|\leq 2R_1 \sqrt{\hbar}} \left|f\left(\frac{w_1-q}{\sqrt{\hbar}}\right) f\left(\frac{u_1-q}{\sqrt{\hbar}}\right)\right|^2 \right)^\frac{1}{2}\\
	&
	\qquad \hbar^{\frac{3}{2}} \int \mathrm{d}{\tilde{q}_2} |f(\tilde{q}_2)|^2 \left(\int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big| (u_1;w_1) \right]^2   \rchi_{|w_1-q|\leq R_1 \sqrt{\hbar}}  \right)^\frac{1}{2}\\
	&
	\leq C \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{L^1} \norm{\nabla V}_{L^\infty}  \hbar^{\frac{3}{2}} \left(\hbar^3  \int \mathrm{d}\tilde{w}_1\mathrm{d}\tilde{u}_1\, \rchi_{|\tilde{w}_1-\tilde{u}_1|\leq \hbar^{\alpha_1 \jinyeop{-} \frac{1}{2}}}  \rchi_{|\tilde{w}_1-\tilde{u}_1|\leq 2R_1} \left|f(\tilde{w}_1) f(\tilde{u}_1) \right|^2 \right)^\frac{1}{2}\\
	&
	\qquad \int\mathrm{d}{q} \left(\int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t} \big| (u_1;w_1) \right]^2  \rchi_{|w_1-q|\leq R_1 \sqrt{\hbar}}   \right)^\frac{1}{2}\\
	&
	\leq C \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{L^1}\norm{\nabla V}_{L^\infty} \hbar^3 \left( \int \mathrm{d}\tilde{w}_1\mathrm{d}\tilde{u}_1\, \rchi_{|\tilde{w}_1-\tilde{u}_1|\leq \hbar^{\alpha_1 \jinyeop{-} \frac{1}{2}}}  \left|f(\tilde{w}_1) f(\tilde{u}_1) \right|^2 \right)^\frac{1}{2}\\
	&
	\qquad \hbar^{ \frac{3}{2}}\int \mathrm{d}{\tilde{q}_1} \rchi_{|\tilde{q}_1|\leq R_1}
	\left(\int \mathrm{d}w_1\mathrm{d}u_1\, \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2  \right)^\frac{1}{2},
\end{align*}
which together with
\begin{align*}
	&\int \mathrm{d}\widetilde{w}_1\, |f(\widetilde{w}_1)|^2 \int \mathrm{d}\widetilde{u}_1 \rchi_{|\widetilde{w}_1-\widetilde{u}_1| \leq \hbar^{\alpha_1-\frac{1}{2}}}  |f(\widetilde{u}_1)|^2\leq \norm{f}_{L^\infty(\mathbb{R}^3)}^2 \norm{f}_{L^2(\mathbb{R}^3)}^2 \hbar^{3(\alpha_1-\frac{1}{2})},
\end{align*}
implies immediately that
\begin{equation*}
	\text{I}_{\mathrm{m}} \leq C \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{L^1} \norm{\nabla V}_{L^\infty}  \hbar^{3+{\frac{3}{2}}(\alpha_1 \jinyeop{-} \frac{1}{2}) + \frac{3}{2}} \left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2  \right)^\frac{1}{2}.
\end{equation*}
	
To balance the order between $\text{I}_{\mathrm{m}}$ and $\text{J}_{\mathrm{m}}$, $s$ is chosen to be
\[
	s = {\left\lceil \frac{3(\alpha_1 \jinyeop{-} \frac{1}{2})}{2(1-\alpha_1)} \right\rceil},
\]
for $\alpha_1 \in \left[0,1\right)$. Therefore, we have
\begin{equation*}
	\begin{aligned}
		\bigg|&\int \mathrm{d}q \mathrm{d}p\, \ \varphi(q) \phi(p) \nabla_{p} \cdot \mathcal{R}_\mathrm{m} \bigg| \leq \text{I}_{\mathrm{m}} +\text{J}_{\mathrm{m}} \\
		&
		\leq C  \norm{\varphi}_{L^\infty} \norm{\nabla \phi}_{W^{s,\infty}} \norm{\nabla V}_{L^\infty}  \hbar^{3+ {\frac{3}{2}}(\alpha_1 \jinyeop{-} \frac{1}{2}) + \frac{3}{2}} \left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2   \right)^\frac{1}{2},
	\end{aligned}
\end{equation*}
and we obtained the desired result.
\end{proof}

Next, we want to bound the term with the `mixed'-norm, i.e.,
\begin{equation}\label{eq_mixnorm_0}
	\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2.
\end{equation}
The following proposition provides the estimate of \eqref{eq_mixnorm_0}:

\begin{Proposition}\label{prop:mixed_norm}
Let $\gamma^{(k)}_{N,t}$ be $k$-particle reduced density matrix associated with $\Psi_{N,t}$, $\omega_{N,t}$ be the solution of the Hartree-Fock equation in \eqref{eq:hartree-fock}. Suppose the assumption for Theorem \ref{thm:main_ben} holds. Then the following inequalities hold for all $t\in \bR$:
	
\begin{equation}\label{eq:theorem_HS}
	\norm{\gamma^{(1)}_{N,t}- \omega_{N,t}}_{\mathrm{HS}} \leq C  e^{c |t|},
\end{equation}
and
\begin{equation}\label{eq:theorem_trace}
	\trnorm{\gamma^{(1)}_{N,t}- \omega_{N,t} }\leq C \sqrt{N} e^{c |t|}.
\end{equation}
Furthermore, it holds that
\begin{equation}\label{eq:mixed_norm_0}
	\left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \omega_{N,t} \otimes \omega_{N,t}  \big| \right]^2(u_1;w_1)   \right)^\frac{1}{2} \leq   C N e^{c |t|},
\end{equation}
where we denote the partial trace $\Tr^{(1)} |\omega_{N,t}(x_1, \cdot\ ;\ y_1, \cdot)| := \int \mathrm{d}{x_2} |\omega_{N,t} (x_1, x_2;y_1, x_2)|$ 
\end{Proposition}
\label{sec:mixednorm}

The proof of Proposition \ref{prop:mixed_norm} requires the following results from \cite{benedikter2014mean}, namely:

\begin{Lemma}[Lemma 3.1 of \cite{benedikter2014mean}]\label{lem:bound_O}
Let $\dd{\Gamma}(O)$ be the second quantization of any bounded operator $O$ on $L^2(\bR^3)$, i.e.
\[
	\dd{\Gamma}(O) :=  \int \mathrm{d}x\mathrm{d}y\ O(x;y) a^*_x a_y.
\]
For any $\Psi \in \mathcal{F}_a$, the following inequalities hold
\begin{equation}\label{eq:bound_O_1}
	\norm{\dd{\Gamma(O)\Psi}} \leq \norm{O} \norm{\mathcal{N} \Psi }.
\end{equation}
If furthermore $O$ is a Hilbert-Schmidt operator, we have the following bounds:
\begin{align}
	\norm{\dd{\Gamma (O) \Psi}} &\leq \norm{O}_{\mathrm{HS}} \norm{\mathcal{N}^{1/2} \Psi },\label{eq:bound_O_2} \\
	\norm{\int \mathrm{d}{x} \mathrm{d}{y} O(x;y) a_x a_y \Psi } & \leq  \norm{O}_{\mathrm{HS}} \norm{\mathcal{N}^{1/2} \Psi },\label{eq:bound_O_3} \\
	\norm{\int \mathrm{d}{x} \mathrm{d}{y} O(x;y) a_x^* a_y^* \Psi } & \leq 2 \norm{O}_{\mathrm{HS}}\norm{(\mathcal{N}+1)^{1/2} \Psi }.\label{eq:bound_O_4}
\end{align}
Finally, if $O$ is a trace class operator, we obtain
\begin{align}
	\norm{\dd{\Gamma (O) \Psi}} &\leq 2  \norm{O}_{\Tr},\label{eq:bound_O_5} \\
	\norm{\int \mathrm{d}{x} \mathrm{d}{y} O(x;y) a_x a_y \Psi } &\leq 2  \norm{O}_{\Tr},\label{eq:bound_O_6} \\
	\norm{\int \mathrm{d}{x} \mathrm{d}{y} O(x;y) a_x^* a_y^* \Psi } & \leq 2  \norm{O}_{\Tr}\label{eq:bound_O_7},
\end{align}
where $\norm{O}_{\Tr} := \Tr |O| = \Tr \sqrt{O^*O}$.
\end{Lemma}

\begin{Lemma}[Proposition 3.4 of \cite{benedikter2014mean}]\label{lem:propagation}
Suppose the assumption for Theorem \ref{thm:main_ben} holds. Then, there exist constants $K,c>0$ depending only on potential $V$ such that
\begin{align*}
	\sup_{p\in\mathbb{R}^{3}}\frac{1}{1+|p|}\Tr|[\omega_{N,t},e^{\mathrm{i} p\cdot x}]| & \leq KN\hbar e^{c|t|}\\
	\Tr|[\omega_{N,t},\hbar\nabla]| & \leq KN\hbar e^{c|t|}
\end{align*}
for all $p \in \bR^3$ and $t\in \bR$.
\end{Lemma}

\begin{Lemma}[Theorem 3.2 of \cite{benedikter2014mean}]\label{lem:est_U}
Let $\mathcal{U}_{N}(t;s)$ be the quantum fluctuation dynamics defined in \eqref{eq:flunctuation} and $\mathcal{N}$ be the number operator. If the assumptions in Lemma \ref{lem:propagation} hold. Then for $\xi_N\in\mathcal{F}_a$ with $\langle \xi_N , \mathcal{N}^k \xi_N \rangle \leq C$ for any $k \geq 1$, we have the following inequality:
\begin{equation}
	\norm{(\mathcal{N}+1)^k \, \mathcal{U}_{N}(t;0)\xi_N} \leq   C_t  ,
\end{equation}
where $C_t := K e^{c|t|} $ is a positive constant depending on $t \in \bR$, $k$ and potential $V$.
\end{Lemma}

\begin{Remark}
    {Here in this paper we only need the result for initial data $\xi_N=\Omega$.}
\end{Remark}

Now, we are ready to provide the proof of Proposition \ref{prop:mixed_norm}.

\begin{proof}[Proof of Proposition \ref{prop:mixed_norm}]

The proof of the inequalities \eqref{eq:theorem_HS} and \eqref{eq:theorem_trace} follows by modifying Theorem 2.1 of \cite{benedikter2014mean}. In particular, from equation (4.3) in \cite{benedikter2014mean}, we obtain
\begin{align*}
	\hsnorm{\gamma^{(1)}_{N,t} - \omega_{N,t}}  &\leq C  \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N},\\
	\trnorm{ \gamma^{(1)}_{N,t} - \omega_{N,t} } &\leq  C  \sqrt{N} \norm{ \mathcal{N}  \mathcal{U}_N(t;0) \xi_N},
\end{align*}
by choosing the appropriate operator $O$ as discussed in \cite{benedikter2014mean}. Our results for \eqref{eq:theorem_HS} and \eqref{eq:theorem_trace} are obtained by applying Lemma \ref{lem:est_U} and taking the assumption that  $\norm{(\mathcal{N}+1)\xi_N}\leq C$.

Therefore, it remains to prove for \eqref{eq:mixed_norm_0}. As remarked previously, the trace norm and Hilbert-Schmidt norm of the difference between $\gamma_{N,t}^{(k)}$ and $\omega_{N,t}^{(k)}$ are obtained separately with the help of Wick's theorem for $k \geq 2$ in \cite{benedikter2014mean}. For our term, however, we do not directly use Wick's theorem to compute \eqref{eq:mixed_norm_0} as each terms requires similar but still unique method when taking the estimation.

Simplifying the notation $\mathcal{R}_{t} := \mathcal{R}_{\mathcal{V}_{N,t}}$, where $\mathcal{R}_{\mathcal{V}_{N,t}}$ is the Bogoliubov transformation given in the Appendix \eqref{eq:exciting_anil_creation}, we have, from the definition of a $2$-particle reduced density matrix and \eqref{eq:exciting_anil_creation}. that
\begin{equation}\label{eq:gamma_omega_0}
	\begin{split}
		&\gamma^{(2)}_{N,t}(x_1,x_2;y_1,y_2) \\
		& = \left< \xi_N, \mathcal{U}^*_N(t;0) \mathcal{R}^*_t a^*_{y_1} a^*_{y_2} a_{x_2} a_{x_1} \mathcal{R}_t    \mathcal{U}_N(t;0) \xi_N \right>\\
		&
		= \left< \xi_N, \mathcal{U}^*_N(t;0) \ \mathcal{R}^*_t a^*_{y_1} \mathcal{R}_t \mathcal{R}^*_t a^*_{y_2} \mathcal{R}_t \mathcal{R}^*_t a_{x_2} \mathcal{R}_t \mathcal{R}^*_t a_{x_1} \mathcal{R}_t  \mathcal{U}_N(t;0) \xi_N \right>\\
		&
		= \bigg<  \xi_N, \mathcal{U}^*_N(t;0)  \left( a^*(\mathrm{u}_{t,{y_1}}) + a(\bar{\mathrm{v}}_{t,{y_1}}) \right) \left( a^*(\mathrm{u}_{t,{y_2}}) + a(\bar{\mathrm{v}}_{t,{y_2}}) \right) \\
		&\qquad \left(a(\mathrm{u}_{t,x_2}) + a^*(\bar{\mathrm{v}}_{t,x_2})\right) \left(a(\mathrm{u}_{t,x_1}) + a^*(\bar{\mathrm{v}}_{t,x_1})\right)    \mathcal{U}_N(t;0)  \xi_N  \big>\\
		&
		= \bigg<  \xi_N, \mathcal{U}^*_N(t;0) \bigg[ a(\bar{\mathrm{v}}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2})a(\mathrm{u}_{t,x_1}) + a(\bar{\mathrm{v}}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,x_1}) \\
		&
		\qquad + a(\bar{\mathrm{v}}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1}) + {  a(\bar{\mathrm{v}}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})}\\
		&
		\qquad + a(\bar{\mathrm{v}}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a(\mathrm{u}_{t,x_2})a(\mathrm{u}_{t,x_1}) + a(\bar{\mathrm{v}}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,x_1})\\
		&
		\qquad + a(\bar{\mathrm{v}}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1}) + a(\bar{\mathrm{v}}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})\\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2})a(\mathrm{u}_{t,x_1}) + a^*(\mathrm{u}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,x_1})\\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1}) + a^*(\mathrm{u}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})\\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,x_1})+ a^*(\mathrm{u}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,x_1})\\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1})a^*(\mathrm{u}_{t,y_2}) a(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})+ a^*(\mathrm{u}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})\bigg] \mathcal{U}_N(t;0)  \xi_N  \bigg>,
	\end{split}
\end{equation}
where we use \eqref{eq:exciting_anil_creation} in the third equality. Note that since  $\left<\bar{\mathrm{v}}_{t,x}, \bar{\mathrm{v}}_{t,y}\right> = {\omega_{N,t}(y;x)}$, it holds that
\begin{equation}\label{eq:gamma_omega_2facotr}
	\begin{split}
		&{ a(\bar{\mathrm{v}}_{t,y_1})a(\bar{\mathrm{v}}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,x_1})} \\&
		= a^*(\bar{\mathrm{v}}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2})a(\bar{\mathrm{v}}_{t,y_2}) + \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2}\right> a(\bar{\mathrm{v}}_{t,y_1}) a^*(\bar{\mathrm{v}}_{t,x_1}) \\
		& \quad - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> a(\bar{\mathrm{v}}_{t,y_1}) a^*(\bar{\mathrm{v}}_{t,x_2}) - \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		= a^*(\bar{\mathrm{v}}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2})a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		\quad - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2}\right> a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) + { \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2}\right> \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1}\right>}\\
		&
		\quad + \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_1}) - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_2}\right>\\
		&
		\quad  - \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		= a^*(\bar{\mathrm{v}}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2})a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		\quad - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2}\right> a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) + {\omega_{N,t} (x_1;y_1) \omega_{N,t} (x_2;y_2)}\\
		&
		\quad + \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_1}) - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_2}\right>\\
		&
		\quad  - \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2}).
	\end{split}
\end{equation}
Therefore, we obtain
\begin{equation}\label{eq:rearranged_CAR_1}
	\begin{split}
		&\gamma^{(2)}_{N,t}(x_1,x_2;y_1,y_2) -{ \omega_{N,t} (x_1; y_1)\omega_{N,t} (x_2; y_2)}  \\
		&
		= \bigg<  \xi_N, \mathcal{U}^*_N(t;0) \bigg[ a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_2}) a(\mathrm{u}_{t,x_2})  \\
		&
		\qquad + a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_1}) a^*(\mathrm{u}_{t,y_2}) a(\mathrm{u}_{t,x_2}) + a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_1}) a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2}) + a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1})a^*(\mathrm{u}_{t,y_2})a(\mathrm{u}_{t,x_2}) \\
		&
		\qquad + a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) a^*(\bar{\mathrm{v}}_{t,x_2}) a^*(\mathrm{u}_{t,y_2}) - a^*(\mathrm{u}_{t,y_1})a(\mathrm{u}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2})  + a^*(\mathrm{u}_{t,y_1}) a(\mathrm{u}_{t,x_1})  a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2}) \\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_2})a(\mathrm{u}_{t,x_2}) + a^*(\bar{\mathrm{v}}_{t,x_1}) a^*(\mathrm{u}_{t,y_1}) a^*(\bar{\mathrm{v}}_{t,x_2})a(\bar{\mathrm{v}}_{t,y_2}) + a^*(\mathrm{u}_{t,y_1})  a(\mathrm{u}_{t,x_1}) a^*(\mathrm{u}_{t,y_2}) a(\mathrm{u}_{t,x_2})\\
		&
		\qquad + a^*(\mathrm{u}_{t,y_1}) a(\mathrm{u}_{t,x_1}) a^*(\mathrm{u}_{t,y_2}) a^*(\bar{\mathrm{v}}_{t,x_2})  + a^*(\mathrm{u}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_1})a^*(\mathrm{u}_{t,y_2})a^*(\bar{\mathrm{v}}_{t,x_2}) +  a^*(\bar{\mathrm{v}}_{t,x_1})a(\bar{\mathrm{v}}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2})a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		\qquad - a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2})  + a^*(\mathrm{u}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_1}) a^*(\mathrm{u}_{t,y_2}) a(\mathrm{u}_{t,x_2}) - a(\mathrm{u}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2})\\
		&
		\qquad
		- \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1} \right> a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_2})
		+ \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1} \right> a(\bar{\mathrm{v}}_{t,y_2}) a(\mathrm{u}_{t,x_2}) \\
		&
		\qquad- \left<\mathrm{u}_{t,x_1}, \mathrm{u}_{t,y_2} \right> a(\bar{\mathrm{v}}_{t,y_1}) a(\mathrm{u}_{t,x_2})
		+ \left<\mathrm{u}_{t,x_1}, \mathrm{u}_{t,y_2} \right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_1})  -  \left<\mathrm{u}_{t,x_1}, \mathrm{u}_{t,y_2} \right> \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_2} \right>\\
		&
		\qquad
		+ \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1} \right> a^*(\mathrm{u}_{t,y_2})  a(\mathrm{u}_{t,x_2})
		+ \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1} \right> a^*(\mathrm{u}_{t,y_2})  a^*(\bar{\mathrm{v}}_{t,x_2})
		\\
		&
		\qquad
		- \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1} \right>  a^*(\mathrm{u}_{t,y_1})a(\mathrm{u}_{t,x_2})
		- \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1} \right> a^*(\mathrm{u}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2})
		\\
		&
		\qquad  - \left<\mathrm{u}_{t,x_1}, \mathrm{u}_{t,y_2} \right> a^*(\mathrm{u}_{t,y_1})a(\mathrm{u}_{t,x_2})
		- \left<\mathrm{u}_{t,x_1},\mathrm{u}_{t,y_2} \right> a^*(\mathrm{u}_{t,y_1})a^*(\bar{\mathrm{v}}_{t,x_2}) \\
		&
		\qquad  - \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2}\right> a^*(\bar{\mathrm{v}}_{t,x_1}) a(\bar{\mathrm{v}}_{t,y_1}) + \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_1}) \\
		&
		\qquad +\left< \bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2} \right> a^*(\mathrm{u}_{t,y_1}) a(\mathrm{u}_{t,x_1}) - \left< \bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_2} \right> a^*(\bar{\mathrm{v}}_{t,x_1}) a^*(\mathrm{u}_{t,y_1})\\
		&
		\qquad   - \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_1}\right> a^*(\bar{\mathrm{v}}_{t,x_2}) a(\bar{\mathrm{v}}_{t,y_2})  -   \left<\bar{\mathrm{v}}_{t,y_2}, \bar{\mathrm{v}}_{t,x_1}\right> \left<\bar{\mathrm{v}}_{t,y_1}, \bar{\mathrm{v}}_{t,x_2}\right> \bigg] \mathcal{U}_N(t;0)  \xi_N  \bigg>,
	\end{split}
\end{equation}
where we used the fact that $ \left<\bar{\mathrm{v}}_{t,x}, {\mathrm{u}}_{t,y} \right> = 0$, $\left<\mathrm{u}_{t,x}, \bar{\mathrm{v}}_{t,y} \right>  = 0$ and CAR.
\begin{align*}
	&\int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)\left( \gamma^{(2)}_{N,t}(z_1,z_2;x_1,x_2) - \omega_{N,t} (z_1; x_1)\omega_{N,t} (z_2; x_2)  \right)\\
	&
	= \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \bigg<  \xi_N, \mathcal{U}^*_N(t;0) \bigg[ a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_2}) a(\mathrm{u}_{t,z_2})  \\
	&
	\qquad + a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2}) + a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,z_2}) + a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1})a^*(\mathrm{u}_{t,x_2})a(\mathrm{u}_{t,z_2}) \\
	&
	\qquad + a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2}) a^*(\mathrm{u}_{t,x_2}) - a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_1})a(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,z_2}) + a^*(\mathrm{u}_{t,x_1}) a(\mathrm{u}_{t,z_1})  a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2}) \\
	&
	\qquad + a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1})a(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,z_2}) + a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2})a(\bar{\mathrm{v}}_{t,x_2}) + a^*(\mathrm{u}_{t,x_1})  a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2})\\
	&
	\qquad + a^*(\mathrm{u}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a^*(\bar{\mathrm{v}}_{t,z_2})  + a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1})a^*(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,z_2}) +  a^*(\bar{\mathrm{v}}_{t,z_1})a(\bar{\mathrm{v}}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2})a(\bar{\mathrm{v}}_{t,x_2})\\
	&
	\qquad - a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2})  + a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2}) - a(\mathrm{u}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2})\\
	&
	\qquad
	- \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right> a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2})
	+ \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a(\bar{\mathrm{v}}_{t,x_2}) a(\mathrm{u}_{t,z_2}) \\
	&
	\qquad- \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2})
	+ \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_1})  -  \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_2} \right>\\
	&
	\qquad
	+ \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_2})  a(\mathrm{u}_{t,z_2})
	+ \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_2})  a^*(\bar{\mathrm{v}}_{t,z_2})  \\
	&
	\qquad
	- \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right>  a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_2})
	- \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2}) \\
	&
	\qquad  - \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_2})
	- \left<\mathrm{u}_{t,z_1},\mathrm{u}_{t,x_2} \right> a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2}) \\
	&
	\qquad  - \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2}\right> a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) + \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1}\right> a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_1}) \\
	&
	\qquad +\left< \bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2} \right> a^*(\mathrm{u}_{t,x_1}) a(\mathrm{u}_{t,z_1}) - \left< \bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2} \right> a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_1})\\
	&
	\qquad   - \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1}\right> a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2})  -  {\left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1}\right> \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_2}\right>} \bigg] \mathcal{U}_N(t;0)  \xi_N  \bigg>\\
	&
	= : \sum_{i=1}^{16} A_i +  \sum_{j=1}^{16} B_j + {C}. \numberthis \label{many_terms}
\end{align*}
Using the fact that $\opnorm{\mathrm{u}_t},\opnorm{\mathrm{v}_t} \leq 1$, $\hsnorm{\mathrm{v}_t} \leq \sqrt{N}$, $\trnorm{\omega_{N,t}} = N$ and the assumption $\norm{\xi_N} \leq 1$, we do the following estimates for the first term from $\{A_i\}_{i=1}^{16}$ and $\{B_i\}_{i=1}^{16}$ separately.
\begin{align*}
	&|A_1|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2  \mathrm{d}z_1 \mathrm{d}z_2\ \big<\xi_N, U^*_N(t;0)    \int  \mathrm{d}\eta_1 \mathrm{d}\eta'_1\ a_{\eta_1} a_{\eta'_1} \mathrm{v}_t(\eta_1;x_1) O_1(x_1;z_1) {\mathrm{u}_t(z_1;\eta'_1)}\\
	&
	\qquad  \int \mathrm{d}\eta_2 \mathrm{d}\eta'_2\ a_{\eta_2} a_{\eta'_2} \mathrm{v}_t(\eta_2;x_2) O_2(x_2; z_2) {\mathrm{u}_t(z_2;\eta'_2)} \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg| \big<\xi_N, U^*_N(t;0)    \int  \mathrm{d}\eta_1 \mathrm{d}\eta'_1\ a_{\eta_1} a_{\eta'_1} \big(\mathrm{v}_t O_1 \mathrm{u}_t\big)(\eta_1;\eta'_1) \int \mathrm{d}\eta_2 \mathrm{d}\eta'_2\ a_{\eta_2} a_{\eta'_2} \big(\mathrm{v}_t O_2 \mathrm{u}_t\big)(\eta_2;\eta'_2) \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg|\int \mathrm{d}\eta_1 \mathrm{d}\eta'_2 \big< \xi_N,  U^*_N(t;0)  a_{\eta_1} a\big(\overline{\mathrm{v}_t O_1 \mathrm{u}_t}(\eta_1;\cdot)\big)  a\big(\overline{\mathrm{v}_t O_2 \mathrm{u}_t}(\cdot;\eta'_2)\big) a_{\eta'_2}  \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg|\int \mathrm{d}\eta_1 \mathrm{d}\eta'_2 \big< a^*\big(\overline{\mathrm{v}_t O_1 \mathrm{u}_t}(\eta_1;\cdot)\big) a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N,   a\big(\overline{\mathrm{v}_t O_2 \mathrm{u}_t}(\cdot;\eta'_2)\big) a_{\eta'_2}  \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	\leq \int \dd{\eta_1}  \norm{ a^*\big(\overline{\mathrm{v}_t O_1 \mathrm{u}_t}(\eta_1;\cdot)\big) a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N} \int \dd{\eta'_2} \norm{ a\big(\overline{\mathrm{v}_t O_2 \mathrm{u}_t}(\cdot;\eta'_2)\big) a_{\eta'_2}  \mathcal{U}_N(t;0) \xi_N } \\
	&
	\leq \int \dd{\eta_1}  \norm{\mathrm{v}_t O_1 \mathrm{u}_t(\eta_1;\cdot)}_2 \norm{a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N} \int \dd{\eta'_2} \norm{ \mathrm{v}_t O_2 \mathrm{u}_t(\cdot;\eta'_2)}_2 \norm{a_{\eta'_2}  \mathcal{U}_N(t;0) \xi_N }\\
	&
	\leq  \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t}  \left(\int \dd{\eta_1} \norm{a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N}^2 \right)^\frac{1}{2} \hsnorm{\mathrm{v}_t O_2 \mathrm{u}_t} \left(\int \dd{\eta'_2} \norm{a_{\eta'_2}  \mathcal{U}_N(t;0) \xi_N }^2 \right)^\frac{1}{2}\\
	&
	\leq \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t} \left(\int \dd{\eta_1}  \left<\xi_N, \mathcal{U}_N^*(t;0)  a_{\eta_1} a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N \right> \right)^\frac{1}{2} \\
	&
	\qquad \hsnorm{\mathrm{v}_t O_2 \mathrm{u}_t}  \left(\int \dd{\eta'_2}  \left<\xi_N, \mathcal{U}_N^*(t;0)  a^*_{\eta'_2} a_{\eta'_2} \mathcal{U}_N(t;0)  \xi_N \right> \right)^\frac{1}{2}\\
	&
	\leq \sqrt{N} \hsnorm{\mathrm{v}_t O_1} \opnorm{\mathrm{u}_t} \hsnorm{\mathrm{v}_t O_2} \opnorm{\mathrm{u}_t}  \norm{\left(\mathcal{N} + 1 \right)^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq \sqrt{N} \opnorm{\mathrm{v}_t} \hsnorm{O_1} \opnorm{\mathrm{u}_t} \hsnorm{\mathrm{v}_t} \opnorm{O_2} \opnorm{\mathrm{u}_t}  \norm{\left(\mathcal{N} + 1 \right)^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2} \norm{\left(\mathcal{N} + 1 \right)^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}
Additionally, we have
\begin{align*}
	&|B_1|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right> a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2})    \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \overline{\omega_{N,t}(x_2;z_1)} \\
	&
	\qquad \int  \mathrm{d}\eta \mathrm{d}\eta'\ a_{\eta} a_{\eta'} \mathrm{v}_t(\eta;x_1) \overline{\mathrm{u}_t(\eta'; z_2)}   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2 \big<  \xi_N, \mathcal{U}^*_N(t;0) , \int  \mathrm{d}\eta \mathrm{d}\eta'\ a_{\eta}    a_{\eta'} \\
	&
	\qquad \mathrm{v}_t(\eta;x_1) O_1(x_1;z_1) \omega_{N,t}(z_1;x_2)  O_2(x_2;z_2) {\mathrm{u}_t(z_2;\eta')}   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg|  \int  \mathrm{d}\eta \mathrm{d}\eta' \big<  \xi_N, \mathcal{U}^*_N(t;0) , a_{\eta}    a_{\eta'}   \big(\mathrm{v}_t O_1 \omega_{N,t} O_2 \mathrm{u}_t\big)(\eta;\eta')   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{v}_t O_1 \omega_{N,t} O_2 \mathrm{u}_t} \norm{\xi_N} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq  \opnorm{\mathrm{v}_t} \hsnorm{O_1} \opnorm{\omega_{N,t}} \opnorm{O_2} \opnorm{\mathrm{u}_t} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq  \hsnorm{O_1} \opnorm{O_2} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}


The estimates for the rest of the terms can be done with similar steps, some of which are given in Appendix \ref{sec:mixednorm-rest}. After getting the bound of each terms, we have obtained the following estimates:
\begin{equation}
	\begin{aligned}
		\left|\sum_{i=1}^{16} A_i \right| &\leq  N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}, \\
		\left|\sum_{j=1}^{16} B_i \right| &\leq  \hsnorm{O_1} \opnorm{O_2} \bigg(N \norm{ (\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N} + \sqrt{N}  \norm{ (\mathcal{N}+1) \mathcal{U}_N(t;0)  \xi_N}   \bigg)\\
		&
		\leq   N  \hsnorm{O_1} \opnorm{O_2} \norm{ (\mathcal{N}+1) \mathcal{U}_N(t;0)  \xi_N}.
	\end{aligned}
\end{equation}


Lastly, the final term is estimated as follows:
\begin{align*}
	&|C|\\
	& = \bigg|\int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0)\left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1}\right> \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_2}\right> \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg|\int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) \omega_{N,t}(z_1;x_2) \omega_{N,t}(z_2;x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg|\int  \dd{x_2}   \big<  \xi_N, \mathcal{U}^*_N(t;0) \big( O_1 \omega_{N,t} O_2 \omega_{N,t} \big) (x_2;x_2) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \bigg<  \xi_N, \mathcal{U}^*_N(t;0) \left( \int  \dd{x_2}  \big( \bar{\omega}_{N,t} \overline{O^*_1} \bar{\omega}_{N,t} \overline{O^*_2} \big) (x_2;x_2)\right) \mathcal{U}_N(t;0)  \xi_N \bigg> \bigg|\\
	&
	\leq \trnorm{O_1 \omega_{N,t} O_2 \omega_{N,t}} \big|\big< \xi_N, \mathcal{U}^*_N(t;0) \mathcal{U}_N(t;0) \xi_N \big> \big|\\
	&
	\leq \hsnorm{O_1 \omega_{N,t}} \hsnorm{O_2 \omega_{N,t}}\\
	&
	\leq \hsnorm{O_1} \opnorm{\omega_{N,t}} \opnorm{O_2} \hsnorm{\omega_{N,t}}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2}.
\end{align*}

As a summary, we have
\begin{equation}\label{eq:estimated_mixed}
	\begin{split}
		\left|\Tr O (\gamma_{N,t}^{(2)}- \omega_{N,t} \otimes \omega_{N,t} )\right| & \leq \bigg|\sum_{i=1}^{16} A_i\bigg| + \bigg|\sum_{j=1}^{16} B_j\bigg| + |C|\\
		&
		\leq N \hsnorm{O_1} \opnorm{O_2} \norm{(\mathcal{N}+1) \mathcal{U}_N(t;0) \xi_N },
	\end{split}
\end{equation}
which implies that, for $O_1$ and $O_2$ being Hilbert-Schmidt and trace class operators, we get
\begin{equation}
	\begin{aligned}
		&\left(\int \mathrm{d}x_1\mathrm{d}y_1\ \left[ \int \dd{x_2} \left|\gamma_{N,t}^{(2)}(x_1,x_2;y_1,x_2)  - \omega_{N,t}(x_1;y_1)\omega_{N,t}(x_2;x_2) \right| \right]^2  \right)^\frac{1}{2}\\
		&
		\qquad \leq  N  \norm{(\mathcal{N}+1) \mathcal{U}_N(t;0) \xi_N } .
	\end{aligned}
\end{equation}
Applying Lemma \ref{lem:est_U}, we obtain the inequalities in Proposition \ref{prop:mixed_norm} as desired.
\end{proof}

Finally, we have the following estimate for the mixed-norm. Since it is one of the main contributions of this paper, we write it as a theorem.

\begin{Theorem}\label{thm:mixed_norm_factorized}
	Suppose the assumptions given in Proposition \ref{prop:mixed_norm} hold. Then, we have the following  estimate
	\begin{equation}\label{eq:col_mixed}
		\left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \gamma^{(1)}_{N,t} \otimes \gamma^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2  \right)^\frac{1}{2} \leq  C_t N,
	\end{equation}
	where the constant $C_t$ depends on potential $V$ and time $t$.
\end{Theorem}


\begin{proof}[Proof of Theorem \ref{thm:mixed_norm_factorized}]
	Inserting the intermediate terms
	\begin{equation}\label{eq:insert}
	\begin{aligned}
		&\gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)  - \gamma_{N,t}^{(1)}(u_1;w_1)\gamma_{N,t}^{(1)}(u_2;w_2) \\
		&
		=  \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2) - \omega_{N,t}(u_1;w_1) \omega_{N,t}(u_2;w_2)\\
		&
		\quad + \big[ \omega_{N,t}(u_1;w_1)  - \gamma^{(1)}_{N,t}(u_1;w_1)\big] \omega_{N,t}(u_2;w_2)\\
		&
		\quad + \gamma^{(1)}_{N,t}(u_1;w_1) \big[\omega_{N,t}(u_2;w_2) -  \gamma_{N,t}^{(1)}(u_2;w_2)\big]\\
		&
		=: T_1 + T_2 +T_3,
	\end{aligned}
\end{equation}
the estimate in \eqref{eq:col_mixed} is then reduced into the estimates of the following terms.
	\begin{itemize}
		\item $\left(\int \mathrm{d}w_1\mathrm{d}u_1\ \left[ \Tr^{(1)} \big|\gamma_{N,t}^{(2)}  - \omega^{(1)}_{N,t} \otimes \omega^{(1)}_{N,t}  \big| (u_1;w_1) \right]^2   \right)^\frac{1}{2}$,
		
		\item $ \hsnorm{\omega_{N,t} -  \gamma_{N,t}^{(1)}} \trnorm{\omega_{N,t}}$,
		
		\item $\hsnorm{\gamma_{N,t}^{(1)}} \trnorm{\omega_{N,t} -  \gamma_{N,t}^{(1)}}$.
	\end{itemize}
Proposition \ref{prop:mixed_norm} implies immediately \eqref{eq:col_mixed} considering the fact that $\trnorm{\omega_{N,t}} \leq N$.
\end{proof}

\begin{Corollary}\label{cor:est_mean-field_osc}
	Let $\varphi, \phi \in C_0^\infty(\bR^3)$ are test functions, then the following inequality holds:
	\begin{equation}
			\bigg|\int \mathrm{d}q \mathrm{d}p\, \varphi(q)\phi(p)  \nabla_{p} \cdot \mathcal{R}_\mathrm{m}(q,p) \bigg| \leq {C} \hbar^{\frac{3}{2}(\alpha_1 \jinyeop{-} \frac{1}{2}) + \frac{3}{2}}
	\end{equation}
where $\frac{1}{2} < \alpha_1 < 1$, $C$ depends on $\phi, \varphi, f$.
\end{Corollary}





\section{Estimate for semi-classical residue}\label{sec:mean_field_residue}


In this section, we will estimate the semi-classical residual term under the assumption with $V\in W^{2,\infty}(\bR^3)$ in Proposition \ref{prop_est_R_1}, with which give us the insight to compare the rate between semi-classical and mean-field residuals.
{
\begin{Proposition}\label{prop_est_R_1} Assume that $V\in W^{2,\infty}$ and $V(x) = V(-x)$ and let $\phi, \varphi$ are test functions, then the following inequality holds:
	\begin{equation}
		\begin{aligned}
			\bigg|\int \mathrm{d}q \mathrm{d}p\, \varphi(q) \phi(p) \nabla_{p} \cdot \mathcal{R}_\mathrm{s}(q,p) \bigg| 	\leq C \hbar^{\frac{1}{2}+3(\alpha_2 - 1)} ,
		\end{aligned}
	\end{equation}
where $\frac{1}{2} < \alpha_2 < 1$, and $C$ depends on $\varphi, \phi, f$.
\end{Proposition}

\begin{proof}
First, recall that
	\begin{equation}
		\begin{split}
			\mathcal{R}_\mathrm{s} := & \dfrac{1}{(2\pi)^3}  \int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2 \mathrm{d}{{q_2}} \mathrm{d}p_2    \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2} \\
			&\qquad \left[\int_0^1 \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big) - \nabla V(q-{{q_2}}) \right]  \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2).
		\end{split}
	\end{equation}
Since $\phi(q)$, $\varphi(p)$ are test functions, we estimate	
	\begin{align*}
		&\bigg|\int \mathrm{d}q \mathrm{d}p\, \phi(q)\varphi(p)  \nabla_{p} \cdot \mathcal{R}_\mathrm{s}(q,p) \bigg| \\
		&
		=  \frac{1}{ (2\pi)^3} \bigg| \int (\mathrm{d}q \mathrm{d}p)^{\otimes 2}\ \phi(q) \nabla_{p} \varphi(p) \cdot \int \mathrm{d}w_1 \mathrm{d}u_1 \mathrm{d}w_2\mathrm{d}u_2  \left(  f^\hbar_{q,p}(w)  \overline{f^\hbar_{q,p}(u)} \right)^{\otimes 2} \\
		&
		\qquad \left[\int_0^1 \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big) - \nabla V(q-{{q_2}}) \right]  \gamma_{N,t}^{(2)}(u_1,u_2;w_1,w_2)\bigg|\\
		&
		= \dfrac{1}{(2\pi\hbar)^3} \bigg| \int (\mathrm{d}q)^{\otimes 2}\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla_{p} \varphi(p) f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
			&
		\qquad e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)} \int\mathrm{d}u_2 \mathrm{d}p_2\, e^{\frac{\mathrm{i}}{\hbar}p_2 \cdot (w_2-u_2)}f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) f\left(\frac{u_2 -{{q_2}}}{\sqrt{\hbar}}\right) \\
		&\qquad\left[\int_0^1 \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big) - \nabla V(q-{{q_2}}) \right]
		 \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|\\
		&
		= \bigg| \int (\mathrm{d}q)^{\otimes 2}\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla_{p} \varphi(p) f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
		&
		\qquad e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)} \left|f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) \right|^2  \left[\int_0^1 \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big) - \nabla V(q-{{q_2}}) \right] \\
		&
		\qquad \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|,
	\end{align*}
where we applied the fact that $ (2\pi \hbar)^3 \delta_x(y) = \int  e^{\frac{\mathrm{i}}{\hbar} p \cdot (x-y)} \dd{p}$. Then, inserting $\pm \nabla V(q-w_2)$ and we have
	\begin{align*}
		&
		\leqslant    \bigg| \int (\mathrm{d}q)^{\otimes 2}\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla \varphi(p) \cdot f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
		&
		\qquad e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)} \left|f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) \right|^2  \left[ \int_0^1 \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big)  - \nabla V(q - w_2) \right] \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|\\
		&
		\quad +  \bigg| \int (\mathrm{d}q)^{\otimes 2}\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla \varphi(p) \cdot f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
		&
		\qquad e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)} \left|f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) \right|^2  \left[\nabla V(q - w_2 )  - \nabla V(q - {{q_2}}) \right] \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|\\
		& =: I_\mathrm{s} + J_\mathrm{s}
	\end{align*}
where we used integration by part in the second to last equality.\\
	
Before advancing, recalling \eqref{eq:estimate_oscillation_0}, we split the integral and obtain the following estimate, $\forall\alpha_2\in(\frac{1}{2},1)$,
	\begin{equation}\label{momentum_split}
		\begin{aligned}
			\left|\int \dd{p} \nabla \varphi(p) e^{\frac{i}{\hbar}p\cdot(w-u)}\right|
			&=\left|\int \dd{p}  (\rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha}+ \rchi_{(w_1-u_1)\in (\Omega_\hbar^\alpha)^c}) \nabla \varphi(p) e^{\frac{i}{\hbar}p\cdot(w-u)}\right|\\
			&\leq \tilde{C} \left(\rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s}\right),
		\end{aligned}
	\end{equation}
where $\tilde{C}$ depends on $\norm{\phi}_{W^{s+1,\infty}}$ and $\supp \phi$.\\
	
Now we want to estimate the term $I_\mathrm{s}$ and $J_\mathrm{s}$ separately. We begin by estimating $I_\mathrm{s}$,
	\begin{align*}
		I_\mathrm{s} &
		= \hbar^{\frac{3}{2}} \bigg| \int \mathrm{d}q\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla \varphi(p) f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)}\\
		&
		\qquad \left( \int \mathrm{d}\tilde{q}_2\left|f\left(\tilde{q}_2\right) \right|^2 \right) \left[ \int_0^1  \mathrm{d}s\ \nabla V\big(su_1+ (1-s)w_1 - w_2 \big)  - \nabla V(q - w_2) \right]\\
		& \qquad \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\Big|.
	\end{align*}
Using \eqref{momentum_split}, we have
	\begin{align*}
		I_\mathrm{s} &{\leq}  \norm{D^2 V}_{L^\infty} \tilde{C} \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2 \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left( |u_1-q|+|w_1-q| \right) \big|  \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\big|\\
		&
		\leq C \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left( |u_1-q|+|w_1-q| \right) \int \mathrm{d}w_2 \big| \gamma_{N,t}^{(2)}( w_1,w_2; u_1, w_2)\big|\\
		&
		=  C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1  \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left( |u_1-q|+|w_1-q| \right) \int \mathrm{d}w_2 \big|  \left<a_{w_2}a_{w_1} \psi_{N,t},  a_{w_2}  a_{u_1} \psi_{N,t} \right>\big|\\
		&
		\leq   C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left( |u_1-q|+|w_1-q| \right) \int \mathrm{d}w_2 \norm{a_{w_2}a_{w_1} \psi_{N,t}} \norm{  a_{w_2}  a_{u_1} \psi_{N,t}}\\
		&
		\leq C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left( |u_1-q|+|w_1-q| \right)  \left(\int \mathrm{d}w_2 \norm{ a_{w_2}a_{w_1} \psi_{N,t}}^2\right)^\frac{1}{2} \left(\int \mathrm{d}w_2 \norm{  a_{w_2}  a_{u_1} \psi_{N,t}}^2\right)^\frac{1}{2}\\
		& =: C  \bigg[ i_{\mathrm{s},1} + i_{\mathrm{s},2}\bigg],
	\end{align*}
where we use $i_{s,1}$ to be the term with $ \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha}$, and $i_{s,2}$ to be the other one.


{
Due to the symmetric property , we can reduce the estimate for $i_{\mathrm{s},1}$ into the following 	
	\begin{align}\nonumber
		i_{\mathrm{s},1} &  	\leq 2C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|
		\cdot |u_1-q| \int \mathrm{d}w_2 \norm{ a_{w_2}a_{w_1} \psi_{N,t}}^2\\
	\nonumber	&  	\leq 2C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|
		\cdot |u_1-q|\left<\psi_{N,t}, a^*_{w_1} \mathcal{N} a_{w_1} \psi_{N,t}\right> \\
	\nonumber	&\leq C\hbar^{\frac{3}{2}} \norm{f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)
			\cdot |u_1-q|}_{L^\infty} |\Omega_\hbar^\alpha|  \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1\bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right)\bigg|\left<\psi_{N,t}, a^*_{w_1} \mathcal{N} a_{w_1} \psi_{N,t}\right>
		\\
	\nonumber	&\leq C \hbar^{\frac{3}{2}} R_1\hbar^{\frac{1}{2}} \hbar^{3\alpha_2} \hbar^{\frac{3}{2}} \int \mathrm{d}w_1\left<\psi_{N,t}, a^*_{w_1} \mathcal{N} a_{w_1} \psi_{N,t}\right>\\
	\label{is1}	&\leq C \hbar^{3\alpha_2+3+\frac{1}{2}-6}
	\end{align}
This above estimate shows that the error is of order $\hbar^{\frac{1}{2}-}$ which should be the optimal rate for semi-classical limit.
{Jinyeop: The optimal rate for semi-classical limit expected to be $\hbar$.}
}

Using similar steps with $i_{\mathrm{s},1}$, noticing that $\phi(q)$ has compact support, one obtains the estimate for the term $i_{\mathrm{s},2}$,
\begin{align*}
	i_{\mathrm{s},2} &  	\leq 2C  \hbar^{\frac{3}{2}} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \hbar^{(1-\alpha_2)s} \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|
	\cdot |u_1-q| \int \mathrm{d}w_2 \norm{ a_{w_2}a_{w_1} \psi_{N,t}}^2\\
	 &  	\leq 2C  \hbar^{\frac{3}{2}+\frac{3}{2}+\frac{1}{2}} \int \ \mathrm{d}\tilde u \rchi_{|\tilde u|\leq R_1}|f(\tilde u)| |\tilde u|\int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1    \hbar^{(1-\alpha_2)s} \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) \bigg|\int \mathrm{d}w_2 \norm{ a_{w_2}a_{w_1} \psi_{N,t}}^2\\
	&\leq C \hbar^{\frac{3}{2}+\frac{3}{2}+\frac{1}{2}}  \hbar^{(1-\alpha_2)s} \hbar^{\frac{3}{2}}\hbar^{-6}\leq C \hbar^{(1-\alpha_2)s-1}
\end{align*}
To balance the order between $i_{\mathrm{s},1}$ and $i_{\mathrm{s},2}$, the term $s$ is chosen to be
	\[
	s = \left\lceil \frac{3(\alpha_2-\frac{1}{2})}{1-\alpha_2}\right\rceil,
	\]
	where $\alpha_2 \in (\frac{1}{2},1)$. Therefore, we have
	\begin{equation}\label{R1_I}
		I_\mathrm{s} \leq C \hbar^{\frac{1}{2}+3(\alpha_2-1)}.
	\end{equation}

Now, to estimate $J_\mathrm{s}$, we recall the estimate in \eqref{momentum_split} and obtain
\begin{align*}
		J_\mathrm{s} =&\bigg| \int (\mathrm{d}q)^{\otimes 2}\mathrm{d}p \mathrm{d}w_1 \mathrm{d}u_1  \mathrm{d}w_2\ \phi(q) \nabla \varphi(p) \cdot f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
		&
		\qquad e^{\frac{\mathrm{i}}{\hbar}p \cdot (w_1-u_1)} \left|f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) \right|^2  \left[\nabla V(q - w_2 )  - \nabla V(q - {{q_2}}) \right] \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|\\
		\leq & \norm{D^2V}_{L^\infty}\bigg| \int (\mathrm{d}q)^{\otimes 2}|\phi(q)|\int \mathrm{d}w_1 \mathrm{d}u_1 \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\\
		&
		\qquad \int \mathrm{d}w_2 \left|f\left(\frac{w_2 -{{q_2}}}{\sqrt{\hbar}}\right) \right|^2  |w_2-{{q_2}}| \gamma_{N,t}^{(2)}(u_1,w_2;w_1,w_2)\bigg|\\
			&
		\leq C \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \int \mathrm{d}\tilde q \left|f\left(\tilde q\right) \right|^2  h^{\frac{3}{2}} h^{\frac{1}{2}}|\tilde q| \int \mathrm{d}w_2 \big| \gamma_{N,t}^{(2)}( w_1,w_2; u_1, w_2)\big|\\
		&
		\leq   C  \hbar^{2} \int \mathrm{d}q\ |\phi(q )|\int \mathrm{d}w_1 \mathrm{d}u_1   \left( \rchi_{(w_1-u_1)\in \Omega_\hbar^\alpha} + \hbar^{(1-\alpha_2)s} \right)   \bigg|f\left(\frac{w_1 -q}{\sqrt{\hbar}}\right) f\left(\frac{u_1 -q}{\sqrt{\hbar}}\right)\bigg|   \\
		&
		\qquad \left(\int \mathrm{d}w_2 \norm{ a_{w_2}a_{w_1} \psi_{N,t}}^2\right)^\frac{1}{2} \left(\int \mathrm{d}w_2 \norm{  a_{w_2}  a_{u_1} \psi_{N,t}}^2\right)^\frac{1}{2}\\
		& =: C  \bigg[ j_{\mathrm{s},1} + j_{\mathrm{s},2}\bigg].
	\end{align*}
The estimate for $j_{\mathrm{s},1}$ can be exactly done as in \eqref{is1} for $i_{\mathrm{s},1}$, the same for $j_{\mathrm{s},2}$ as in $i_{\mathrm{s},2}$. Therefore we obtain the same rate for $J_{\mathrm{s}}$ as in \eqref{R1_I} for $I_{\mathrm{s}}$. This completes the proof.
\end{proof}
}

{
\begin{Remark}
The key step in the estimates of semi-classical residue is in \eqref{is1}, with which the computational bugs appeared in \cite{Chen2021JSP,Chen2021AHP} can both be fixed by the same technique.
\end{Remark}
	
\begin{Remark}
The convergence rate of the semi-classical residue $\mathcal{R}_\mathrm{s}$  and mean-field residue $\mathcal{R}_\mathrm{m}$ are of the following orders
\[
	\mathcal{R}_\mathrm{s}\sim \hbar^{\frac{1}{2}-},\qquad \mathcal{R}_\mathrm{m}\sim N^{-\frac{1}{2}}\hbar^{1-}.
\]
Notice that the rates that we obtained for the semi-classical residue should be optimal, while for the mean field residue it is not optimal, nevertheless one can observe that the semi-classical and mean field residue terms are not of the same order in the combined limit argument.
\end{Remark}
}

\section{Proof of Theorem \ref{thm:main_ben}}\label{sec:proof_mainTheorem}
{{
In this section, we prove the main theorem. As have been mentioned in the introduction, we give first the estimate for $\widetilde{\mathcal{R}}$.

\begin{Proposition}\label{prop.tildeR}
Under the assumptions in theorem \ref{thm:main_ben}, the following estimate holds
\begin{align*}
	\norm{\int\mathrm{d}p\,\widetilde{\mathcal{R}}(p,\cdot)}_{L^{\frac{5}{4}}(\mathbb{R}^3)}\leq C\hbar.
\end{align*}
\end{Proposition}

\begin{proof}
Noticing that
\begin{align*}
	\left| \int  \mathrm{d}p \,  \tilde{\mathcal{R}}\right| 
	&
	\leq \hbar  \int \mathrm{d}p \norm{\nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t}}\norm{ a (f^\hbar_{q,p}) \Psi_{N,t} }\\
	&
	\leq   \left[ \hbar^2 \int \mathrm{d}p  \left< \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t},   \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t} \right> \right]^\frac{1}{2} \left[\int  \mathrm{d}p\, m_{N,t}(q,p) \right]^\frac{1}{2}\\
	&
	\leq \left[ \hbar^2 \int \mathrm{d}p  \left< \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t},   \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t} \right> \right]^\frac{1}{2} \varrho_{N,t}^\frac{1}{2}\\
	& 	\leq \left[ \hbar^{\frac{1}{2}} \int \mathrm{d}q \mathrm{d}p  \int  \mathrm{d}w \mathrm{d}u\   \nabla_{q} f\left(\frac{w-q}{\sqrt{\hbar}}\right)\nabla_{q} f\left(\frac{u-q}{\sqrt{\hbar}}\right) e^{\frac{\mathrm{i}}{\hbar}p \cdot (w-u)} \left< \Psi_{N,t},  a^*_w a_u \Psi_{N,t} \right> \right]^\frac{1}{2} \varrho_{N,t}^\frac{1}{2}\\
	& 	\leq \left[  \hbar^{\frac{1}{2}+3} \int \mathrm{d}q \mathrm{d}w\ \hbar^{-1}\left|\nabla f\left(\frac{w-q}{\sqrt{\hbar}}\right) \right|^2 \left< \Psi_{N,t},  a^*_w a_w \Psi_{N,t} \right>\right]^\frac{1}{2} \varrho_{N,t}^\frac{1}{2}\\
	& 	\leq \hbar^{\frac{1}{2}}\|\nabla f\|_2 \varrho_{N,t}^\frac{1}{2},
\end{align*}
which implies
\begin{align*}
	\int \dd{q} \left| \int  \mathrm{d}p\,  \tilde{\mathcal{R}} \right|^\frac{5}{4} 
	\leq \left[ \hbar^2 \int \mathrm{d}q \mathrm{d}p\, \left< \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t},   \nabla_{q} a (f^\hbar_{q,p}) \Psi_{N,t} \right> \right]^\frac{5}{8} \left(\int \dd{q}\varrho_{N,t}^{\frac{5}{3}}\right)^\frac{3}{8}
	\leq \hbar^{\frac{5}{4}} C.
\end{align*}
In the above estimate, we have used the fact that $\|\rho_{N,t}\|_{L^\infty(0,T;L^\frac{5}{3}(\mathbb{R}^3))}\leq C$, which is a direct result from Appendix \ref{sec:apriori}.
\end{proof}


Next, we will show that, for any $T>0$, the sequence $m_{N,t}$ is weakly compact 
and that any accumulation point $m_t$ is exactly the solution of the Vlasov equation.
For this purpose, we need the following lemma.
\begin{Lemma}
Let $m_{N,t}$ be weak solution of the reformulated Schr\"odinger equation \eqref{eq:BBGKY_k1_ben} and $\rho_{N,t}(q):=\int dp\, m_{N,t}(p,q)$.
Then there exists a subsequence of $m_{N,t}$ (without relabeling for convenience) and function $m_t$ such that as $N\rightarrow \infty$
\begin{align}
&m_{N,t}\stackrel{*}{\rightharpoonup} m_t\quad {\rm{in}}\quad L^\infty(0,T;L^s(\mathbb{R}^3\times\mathbb{R}^3)),\quad s\in [1,\infty],\label{lN1}\\
&\nabla V\ast \rho_{N,t}\rightarrow \nabla V\ast\rho_t \quad {\rm{in}}\quad L^r(0,T;L^r(\mathbb{R}^3)),\quad r\in(1,\infty),\label{lN2}
\end{align}
where $\rho_t=\int dp\, m_t$.
\end{Lemma}
\begin{proof}
The estimates in Appendix \ref{sec:apriori} imply
\begin{align}\label{lN3}
\|m_{N,t}\|_{L^\infty(0,T;L^1(\mathbb{R}^3\times\mathbb{R}^3))}
+\|m_{N,t}\|_{L^\infty(0,T;L^\infty(\mathbb{R}^3\times\mathbb{R}^3))}\leq C,
\end{align}
where $C$ appeared in this section denotes a positive constant independent of $N$.
And combining interpolation inequality, we have
\begin{align*}
\|m_{N,t}\|_{L^\infty(0,T;L^s(\mathbb{R}^3\times\mathbb{R}^3))}\leq C,\quad s\in[1,\infty].
\end{align*}
Therefore, \eqref{lN1} is a direct consequence of the above inequality and the moment estimates in Proposition \ref{prop:2nd_moment_finite}.

Furthermore Proposition \ref{prop:2nd_moment_finite}
\begin{align*}
\||p|^2 m_{N,t}\|_{L^\infty(0,T;L^1(\mathbb{R}^3\times\mathbb{R}^3))}\leq C,
\end{align*}
together with \eqref{lN3}, we arrive at
\begin{align*}
\|\rho_{N,t}\|_{L^\infty(0,T;L^s(\mathbb{R}^3))}\leq C, \quad s\in \Big[1,\frac{5}{3} \Big].
\end{align*}
This implies that
\begin{align*}
\rho_{N,t}\stackrel{*}{\rightharpoonup}\rho_t \quad {\rm{in}}\quad L^\infty(0,T;L^s(\mathbb{R}^3)), \quad s\in \Big(1,\frac{5}{3} \Big].
\end{align*}
Owing to $V\in W^{2,\infty}(\mathbb{R}^3)$ and Young's convolution inequality, we have for a.e. $t\in(0,T)$,
\begin{align*}
\|\nabla^2V\ast \rho_{N,t}\|_{L^\infty(\mathbb{R}^3)}
\leq \|\nabla^2 V\|_{L^\infty(\mathbb{R}^3)}\|\rho_{N,t}\|_{L^1(\mathbb{R}^3)}\leq C.
\end{align*}
Similarly, we obtain that
\begin{align}\label{lN4}
\|\nabla V\ast \rho_{N,t}\|_{L^\infty(0,T;W^{1,\infty}(\mathbb{R}^3))}\leq C.
\end{align}
By means of \eqref{eq:BBGKY_k1_ben}, we get
\begin{align*}
\partial_t\rho_{N,t}&=\partial_t\int dp\, m_{N,t}(p,q)\\
&=-\nabla_q\cdot \int dp \,p\,m_{N,t}(p,q)+\nabla_q\cdot \int dp\, \widetilde{\mathcal{R}}.
\end{align*}
It is easy to see
\begin{align*}
\partial_t(\nabla V\ast\rho_{N,t})=-\nabla_q\cdot \Big(\nabla V\otimes_\ast\int dp \,p\,m_{N,t}(p,q) \Big)
+\nabla_q\cdot \Big(\nabla V\otimes_\ast \int dp\, \widetilde{\mathcal{R}} \Big),
\end{align*}
where $(u\otimes_\ast v)_{ij}=u_i\ast v_j$ for $(u,v)\in \mathbb{R}^3\times\mathbb{R}^3$.
Noticing that 
\begin{align*}
\Big\|\int dp \,p\,m_{N,t}(p,q) \Big\|_{L^\infty(0,T;L^s(\mathbb{R}^3))}\leq C,\quad s\in \Big[1,\frac{5}{4} \Big],
\end{align*}
we derive for any test function $\tilde \varphi(q)\in W^{1,3}(\mathbb{R}^3)$ and a.e. $t\in(0,T)$,
\begin{align*}
&\Big|\int_{\mathbb{R}^3}dq\,\nabla_q\cdot \Big(\nabla V\otimes_\ast\int dp \,p\,m_{N,t}(p,q) \Big)\tilde \varphi(q)\Big|\\
\leq &C(V)
\Big\|\int dp\,p\,m_{N,t}(p,q) \Big\|_{L^1(\mathbb{R}^3)}\|\nabla\tilde \varphi(q)\|_{L^{3}(\mathbb{R}^3)}.
\end{align*}
For the second term, applying Proposition \ref{prop.tildeR}, we have
\begin{align*}
	\Big|\int dq\,\nabla_q\cdot \Big(\nabla V\otimes_\ast \int dp\, \widetilde{\mathcal{R}} \Big)\tilde\varphi(q) \Big|\leq C(V)\norm{\int \mathrm{d}p\widetilde{\mathcal{R}}}_{L^{\frac{5}{4}}(\bR^3)}\norm{\nabla\varphi(q)}_{L^{3}(\bR^3)}\leq C\hbar\norm{\nabla\varphi(q)}_{L^{3}(\bR^3)} .
\end{align*}
The estimates above show that 
\begin{align}\label{lN5}
\|\partial_t(\nabla V\ast \rho_{N,t})\|_{L^\infty(0,T;W^{-1,3}(\mathbb{R}^3))}\leq C.
\end{align}
The inequalities \eqref{lN4} and \eqref{lN5} allow us to apply Aubin-Lions lemma to infer that \eqref{lN2}. We mention here that the application of Aubin-Lions lemma is proceeded in a sequence of growing balls, and the convergent subsequence is obtained through diagonal rule.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm:main_ben}]
With the help of \eqref{lN1}, \eqref{lN2},  Proposition \ref{prop.tildeR}, Proposition \ref{prop_est_R_1} and Corollary \ref{cor:est_mean-field_osc},
we can take limit $N\rightarrow \infty$ in the weak formulation of the reformulated Schr\"odinger equation \eqref{eq:BBGKY_k1_ben}.
More precisely, for any $\phi,\varphi\in C_0^\infty(\mathbb{R}^3)$ and $\eta\in C_0^\infty[0,T)$, $m_{N,t}$ satisfies the following equation
\begin{align*}
 \int_0^T&\mathrm{d}t\int \mathrm{d}q\mathrm{d}p\, m_{N,t}(p,q)\big[\partial_t\eta\varphi(q)\phi(p)+p\cdot\eta(t)\nabla_q\varphi(q)\phi(p)
 -{\frac{1}{(2\pi)^3}}\nabla V\ast\rho_{N,t}\eta(t)\varphi(q)\cdot\nabla_p\phi(p) \big]\\
 &+\eta(0)\int \mathrm{d}q\mathrm{d}p\,\varphi(q)\phi(p)m_{M,0}\\
 &=-\int_0^T \mathrm{d}t\,\eta(t)\int \mathrm{d}q\mathrm{d}p\,\varphi(q)\phi(p)\big(\nabla_q\cdot \widetilde{\mathcal{R}}+\nabla_p\cdot \mathcal{R} \big).
\end{align*}
Since the sums and products of functions of the
form $\eta(t)\varphi(q)\phi(p)$ are dense in $C_0^\infty([0,T)\times\mathbb{R}^3\times\mathbb{R}^3)$, we have showed that the limit of the subsequence is a weak solution of the Vlasov equation. On the other hand, the assumption $V\in W^{2,\infty}$ implies that the Vlasov equation has a unique weak solution. Therefore, the whole sequence $m_{N,t}$ converges weakly. Hence the proof of Theorem \ref{thm:main_ben} is completed.
\end{proof}
}
}

\clearpage
\begin{appendices}
\section{Second Quantization}\label{sec:FockSpace}
{The Fock space formalism and some results from Bogoliubov theorem for the proof of this paper are listed in the following.} \footnote{See \cite{nam_qm2} for more pedagogic treatment on the topics.} In particular, as in \cite{Benedikter2016book}, we will introduce the fermionic Fock space over Hilbert spaces as the following direct sum:
		\[
		\mathcal{F}_a := \mathbb{C} \bigoplus_{n \geq 1} L^2_a (\bR^{3n}).
		\]
By convention, we say that the vacuum state, denoted as $\Omega = \{1,0,0,\dots \}$, belongs to $\mathbb{C}$. For all $\Psi = \{\psi^{(n)}\}_{n\in \mathbb{N}} \in \mathcal{F}_a$ and $\psi^{(n)} \in L^2_a (\bR^{3n})$, we define the number of particle operator on the $n$-th sector by $\big(\mathcal{N} \Psi \big)^{(n)} = n \psi^{(n)}$.

As in \cite{Benedikter2016book}, the creation and annihilation operators acting on $\Psi \in \mathcal{F}_a$ is defined as follows: for any $f \in L^2(\bR^3)$
	\begin{align*}
		\big(a^*(f)\Psi \big)^{(n)} (x_1, \dots, x_n) &:= \sum_{j=1}^n \frac{(-1)^j}{\sqrt{n}} f(x_j) \psi^{(n-1)} (x_1,\dots,x_{j-1}, x_{j+1},\dots,x_n ),\\
		\big(a(f)\Psi \big)^{(n)} (x_1, \dots, x_n) &:= \sqrt{n+1} \int \dd{x} \overline{f(x)} \psi^{(n+1)}(x,x_1,\dots,x_n),
	\end{align*}
where $\psi^{(n)} \in L^2(\bR^{3n})$ for any $n\in \mathbb{N}$. Additionally, for convenient purposes, the creation and annihilation operators will be represented by its operator-value distribution, $a^*_x$ and $a_x$ respectively, so that
	\[
	a^*(f) = \int \dd{x} f(x) a_x, \quad a(f) = \int \dd{x} \overline{f(x)} a_x.
	\]
Therefore, the canonical anticommutator relation(CAR) is written as
	\[
	\{a^*_x, a_y\} = \delta_{x=y}, \quad \{a^*_x, a^*_y\} = \{a_x, a_y\} = 0,
	\]
for any $x,y \in \bR^3$.

Observe that for given any $\Psi, \Phi \in \mathcal{F}_a$, it holds that
	\[
	\left< \Psi, \mathcal{N} \Phi \right> = \int \mathrm{d}{x} \left< a_x \Psi, a_x \Phi \right>.
	\]
Therefore, we write the number of particles operator as $\mathcal{N} = \int \dd{x} a^*_x a_x$. Similarly, the integral kernel of the $k$-particle reduced density matrix is written as follows:
	\begin{equation}\label{eq:k_reduced}
		\gamma^{(k)} (x_1, \dots, x_k; y_1, \dots, y_k) = \left< \Psi, a^*_{y_1} \dots a^*_{y_k} a_{x_k} \cdots a_{x_1}  \Psi \right>.
	\end{equation}
Moreover, the Hamiltonian acting on $\Psi \in \mathcal{F}_a$ can be written as
	\begin{equation}\label{eq:fock_Hamil}
		\mathcal{H}_N := \frac{\hbar^2}{2}\int \mathrm{d}{x} \nabla_x a^*_x \nabla_x a_x + \frac{1}{2N} \int \mathrm{d}{x}\mathrm{d}{y}\, V(x-y) a^*_x a^*_y a_y a_x,
	\end{equation}
where $V$ is the interaction potential. We will denote the operator of the kinetic term as
	\begin{equation}\label{eq:fock_kinetic}
		\mathcal{K} = \hbar^2 \int \dd{x} \nabla_x a^*_x \nabla_x a_x.
	\end{equation}

As presented in \cite{benedikter2014mean, solo_lect}, for any $t \geq 0$, there exists a unitary transformation $\mathcal{R}_{\mathcal{V}_{N,t}}: \mathcal{F}_a \to \mathcal{F}_a$ such that
\begin{equation}\label{eq:exciting_anil_creation}
	\begin{split}
		\mathcal{R}^*_{\mathcal{V}_{N,t}} a_x \mathcal{R}_{\mathcal{V}_{N,t}} &= a(\mathrm{u}_{x,t}) + a^*(\bar{\mathrm{v}}_{x,t}),\\
		\mathcal{R}^*_{\mathcal{V}_{N,t}} a_x^* \mathcal{R}_{\mathcal{V}_{N,t}} &= a^*(\mathrm{u}_{x,t}) + a(\bar{\mathrm{v}}_{x,t}),
	\end{split}
\end{equation}
where  $\mathrm{v}_{x,t} := \sum_{j=1}^N \dyad{\overline{\mathrm{e}}_{j,t} }{\mathrm{e}_{j,t}}$ and $\mathrm{u}_{x,t} := \mathds{1} - \sum_{j=1}\dyad{{\mathrm{e}}_{j,t} }{\mathrm{e}_{j,t}}$, for any orthonormal basis $\{\mathrm{e}_{j,t}\}_{j=1}^N \subset L^2(\bR^3)$.

Then, for $t \geq 0$, the solution of the Schr\"odinger equation is given as
\begin{equation}\label{eq:psi_bogo}
	\Psi_{N,t} = e^{-\frac{\mathrm{i}}{\hbar}\mathcal{H}_Nt} \mathcal{R}_{\mathcal{V}_{N,0}}\Omega = \mathcal{R}_{\mathcal{V}_{N,t}}\mathcal{U}_{N}(t;0)  \Omega ,
\end{equation}
where $\mathcal{R}_{\mathcal{V}_{N,t}}$ is a unitary Bogoliubov mapping and $\mathcal{U}_{N}$ is the quantum fluctuation dynamics defined as follows,
\begin{equation}\label{eq:flunctuation}
	\mathcal{U}_{N} (t;s) := R_{\mathcal{V}_{N,t}}^*  e^{-\frac{\mathrm{i}}{\hbar}\mathcal{H}_N(t-s)} R_{\mathcal{V}_{N,s}}.
\end{equation}

\clearpage
\section{\textit{A priori} estimates}\label{sec:apriori}

In this appendix, we present in this section a sequence of estimates from \cite{Chen2021JSP} that will prove useful to our calculation. First, we have the following properties of $k$-particle Husimi measures from \cite[Lemma 2.2]{Chen2021JSP}

\begin{Lemma} \label{prop:kHusimi}
	Let $m^{(k)}_{N,t}$ be the $k$-particle Husimi measure as defined in \eqref{eq:husimi_def_1}. Then, the following properties hold:
	\begin{enumerate}
		\item  $m^{(k)}_{N,t}(q,p,\dots,q_k,p_k)$ is symmetric,
		\item  $\frac{1}{(2\pi)^{3k}} \int (\mathrm{d}q\mathrm{d}p)^{\otimes k} m^{(k)}_{N,t}(q,p,\dots,q_k,p_k) = \frac{N(N-1)\cdots (N-k+1)}{N^k}$,
		\item $\frac{1}{(2\pi \hbar)^{3}} \int \mathrm{d}q_k  \mathrm{d}p_k\ m^{(k)}_{N,t}(q,p,\dots,q_k,p_k) = (N-k+1) m^{(k-1)}_{N,t}(q,p,\dots,q_{k-1},p_{k-1}) $,
		\item $ 0 \leq  m^{(k)}_{N,t}(q,p,\dots,q_k,p_k) \leq 1$ a.e.,
	\end{enumerate}
	where $1 \leq k \leq N$.
\end{Lemma}

From \cite[Lemma 2.6]{Chen2021JSP} and \cite[Proposition 2.3]{Chen2021JSP}, we the following estimate for the kinetic energy as well as the moment estimate of the $1$-particle Husimi measure respectively:

\begin{Lemma}\label{lem:kinetic_finite}
	Assume $V \in W^{1,\infty} (\bR^3)$, then the kinetic energy is bounded as follows:
	\begin{equation}\label{eq:k_kinetic_bounded}
		\left<\Psi_{N,t}, \frac{\mathcal{K}}{N} \Psi_{N,t}\right> \leq \left< \Psi_{N}, \mathcal{K} \Psi_{N} \right> + Ct^2,
	\end{equation}
	where $\mathcal{K}$ is defined in \eqref{eq:fock_kinetic} and the constant $C$ depends on $\norm{\nabla V}_\infty$.
\end{Lemma}

\begin{Proposition} \label{prop:2nd_moment_finite} For $t \geq 0$, we have the following finite moments:
	\begin{equation}\label{eq:2nd_moment_finite_0}
		\int \mathrm{d}q \mathrm{d}p\, (|{q}| + |{p}|^2) m_{N,t}(q,p) \leq C(1+t^3),
	\end{equation}
	where $C>0$ is a constant that depends on initial data $\int \mathrm{d}q \mathrm{d}p\, (|q| + |p|^2) m_{N}(q, p) $.
\end{Proposition}

Next, we will present the oscillation estimate from \cite[Lemma 2.5]{Chen2021JSP} which will be used frequently in our proof:

\begin{Lemma}[Bound on localized number operator]\label{lem:localized} Let $\psi_N \in \mathcal{F}^{(N)}_a$ such that $\norm{\psi_N} = 1$, and $R$ be the radius of a ball such that the volume is $1$. Then we have
	\begin{align*}
		&\int \mathrm{d}q\mathrm{d}x \left< \psi_N ,
		\rchi_{|x-q|\leq \sqrt{\hbar}R}a^*_{x}a_{x} \psi_N  \right>
		\leq C(R)\hbar^{-\frac{3}{2}},
	\end{align*}
	where $\rchi$ is a characteristic function.
\end{Lemma}

\begin{Lemma}[Estimate of oscillation]\label{lem:estimate_oscillation} For $\varphi \in C^\infty_0 (\bR^3)$ and
	\begin{equation}\label{eq:estimate_oscillation_omega}
		\Omega_\hbar^\alpha := \{x \in \bR^3;\ \max_{1\leq j \leq 3} |x_j|\leq \hbar^\alpha \},
	\end{equation}
	it holds for every $\alpha \in (0,1)$, $s \in \bN$, and $x \in \bR^3\backslash \Omega_\hbar^\alpha$,
	\begin{equation}\label{eq:estimate_oscillation_0}
		\left|\int_{\bR^3} \mathrm{d}p\,  e^{\frac{\rm i}{\hbar}p\cdot x} \varphi(p)\right| \leq C \hbar^{(1-\alpha)s},
	\end{equation}
	where $C$ depends on the compact support and the $C^s$-norm of $\varphi$.
\end{Lemma}

\clearpage
\section{Rest of Proof of Proposition \ref{prop:mixed_norm}}\label{sec:mixednorm-rest}

As each of the estimation of the terms in \eqref{many_terms} is slightly different, for readers who needs more detail, the rest of the estimation will be presented here in this appendix:

	
\begin{align*}
	&|A_2|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0)  a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2  \mathrm{d}z_1 \mathrm{d}z_2\ \big<\xi_N, U^*_N(t;0) O_1(x_1;z_1) O_2(x_2; z_2)  \int  \mathrm{d}\eta_1 \mathrm{d}\eta'_1\ a_{\eta_1} a_{\eta'_1} \mathrm{v}_t(\eta_1;x_1) \overline{\mathrm{u}_t(\eta'_1; z_1)}\\
	&
	\qquad  \int \mathrm{d}\eta_2 \mathrm{d}\eta'_2\ a^*_{\eta_2} a_{\eta'_2} \mathrm{u}_t(\eta_2;x_2) \overline{\mathrm{u}_t(\eta'_2;z_2)} \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2  \mathrm{d}z_1 \mathrm{d}z_2\ \big<\xi_N, U^*_N(t;0)   \int  \mathrm{d}\eta_1 \mathrm{d}\eta'_1\ a_{\eta_1} a_{\eta'_1} \mathrm{v}_t(\eta_1;x_1) O_1(x_1;z_1) {\mathrm{u}_t(z_1; \eta'_1)}\\
	&
	\qquad  \int \mathrm{d}\eta_2 \mathrm{d}\eta'_2\ a^*_{\eta_2} a_{\eta'_2} \mathrm{u}_t(\eta_2;x_2)  O_2(x_2; z_2) {\mathrm{u}_t(z_2;\eta'_2)} \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg| \big<\xi_N, U^*_N(t;0)   \int  \mathrm{d}\eta_1 \mathrm{d}\eta'_1\ a_{\eta_1} a_{\eta'_1} \big(\mathrm{v}_t O_1 \mathrm{u}_t\big)(\eta_1; \eta'_1) \int \mathrm{d}\eta_2 \mathrm{d}\eta'_2\ a^*_{\eta_2} a_{\eta'_2}  \big(\mathrm{u}_t O_2 \mathrm{u}_t\big)(\eta_2; \eta'_2) \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	= \bigg|\int \mathrm{d}\eta_1 \big< a^*\big(\overline{\mathrm{v}_t O_1 \mathrm{u}_t}(\eta_1; \eta'_1)\big) a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N,    \dd{\Gamma}(\mathrm{u}_t O_2 \mathrm{u}_t) \mathcal{U}_N(t;0) \xi_N \big> \bigg|\\
	&
	\leq \int \mathrm{d}\eta_1 \norm{\mathrm{v}_t O_1 \mathrm{u}_t}_2 \norm{a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N}  \norm{ \dd{\Gamma}(\mathrm{u}_t O_2 \mathrm{u}_t) \mathcal{U}_N(t;0)  \xi_N }\\
	&
	\leq  \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t} \left(\int \dd{\eta_1}  \left<\xi_N, \mathcal{U}_N^*(t;0)  a_{\eta_1} a^*_{\eta_1} \mathcal{U}_N(t;0)  \xi_N \right> \right)^\frac{1}{2} \opnorm{\mathrm{u}_t O_2 \mathrm{u}_t} \norm{\mathcal{N} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \opnorm{\mathrm{v}_t} \hsnorm{O_1} \opnorm{\mathrm{u}_t} \opnorm{\mathrm{u}_t} \opnorm{O_2} \opnorm{\mathrm{u}_t} \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0)  \xi_N}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq  \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_3|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0)  a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t} \hsnorm{\mathrm{u}_t O_2 \bar{\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \left(\mathcal{N} + 1 \right)  \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}    \norm{\left(\mathcal{N} + 1 \right) \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_4|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1})a^*(\mathrm{u}_{t,x_2})a(\mathrm{u}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \opnorm{\bar{\mathrm{v}}_t \overline{O^*_1} \mathrm{v}_t} \opnorm{\mathrm{u}_t O_2 \mathrm{u}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2  \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\left(\mathcal{N} + 1 \right) \mathcal{U}_N(t;0) \xi_N}.
\end{align*}


\begin{align*}
	&|A_5|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2}) a^*(\mathrm{u}_{t,x_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_1} {\mathrm{v}}_t} \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_2} \bar{\mathrm{u}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \left(\mathcal{N} + 1 \right) \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}\left<\xi_N, \mathcal{U}_N^*(t;0) \left(\mathcal{N} + 1 \right)   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\left(\mathcal{N} + 1 \right) \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_6|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_1})a(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,z_2})    \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \opnorm{\mathrm{u}_t O_1 \mathrm{u}_t} \hsnorm{{\mathrm{v}}_t O_2 {\mathrm{u}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq  N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}


\begin{align*}
	&|A_7|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1}) a(\mathrm{u}_{t,z_1})  a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \opnorm{\mathrm{u}_t O_1 \mathrm{u}_t} \opnorm{{\mathrm{v}}_t \overline{O^*_2} {\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}


\begin{align*}
	&|A_8|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1})a(\bar{\mathrm{v}}_{t,x_2})a(\mathrm{u}_{t,z_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \bar{\mathrm{v}}_t} \hsnorm{{\mathrm{v}}_t O_2 {\mathrm{u}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^\frac{1}{2}\mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_9|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2})a(\bar{\mathrm{v}}_{t,x_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_1} \bar{\mathrm{u}}_t} \opnorm{\bar{\mathrm{v}}_t \overline{O^*_2} {\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{10}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1})  a(\mathrm{u}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \opnorm{\mathrm{u}_t O_1 {\mathrm{u}}_t} \opnorm{\mathrm{u}_t O_2 \mathrm{u}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{11}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1}) a(\bar{\mathrm{v}}_{t,z_1})a^*(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,z_2})  \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \mathrm{u}_t} \hsnorm{\mathrm{u}_t O_2 \bar{\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) (\mathcal{N} + 1 ) \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0)  \left(\mathcal{N} + 1 \right)  \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{(\mathcal{N} + 1 )^\frac{1}{2}\mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{12}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1})a^*(\mathrm{u}_{t,x_2})a^*(\bar{\mathrm{v}}_{t,z_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 {\mathrm{v}}_t} \hsnorm{\mathrm{u}_t O_2 \bar{\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0)  \left(\mathcal{N} + 1 \right) \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0)  \left(\mathcal{N} + 1 \right)   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{(\mathcal{N} + 1 )^\frac{1}{2}\mathcal{U}_N(t;0) \xi_N}.
\end{align*}


\begin{align*}
	&|A_{13}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2})a(\bar{\mathrm{v}}_{t,x_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \opnorm{\bar{\mathrm{v}}_t \overline{O^*_1} {\mathrm{v}}_t} \opnorm{\bar{\mathrm{v}}_t \overline{O^*_2} {\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0)  \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{14}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_1} {\mathrm{v}}_t} \hsnorm{\bar{\mathrm{u}}_t \overline{O^*_2} {\mathrm{v}}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{15}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \bar{\mathrm{v}}_t} \opnorm{{\mathrm{u}}_t O_2 {\mathrm{u}}_t} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N} \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|A_{16}|\\
	& = \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0)  a(\mathrm{u}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2}) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\bar{\mathrm{u}}_t  \overline{O^*_1} \bar{\mathrm{v}}_t} \opnorm{\bar{\mathrm{v}}_t \overline{O^*_1} \mathrm{v}_t}  \left<\xi_N, \mathcal{U}_N^*(t;0) \mathcal{N}^2 \mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2} \left<\xi_N, \mathcal{U}_N^*(t;0) (\mathcal{N} +1)\mathcal{U}_N(t;0) \xi_N\right>^\frac{1}{2}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \left<\xi_N, \mathcal{U}_N^*(t;0)\left(\mathcal{N} + 1 \right)^2   \mathcal{U}_N(t;0) \xi_N\right>\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

Additionally, we have
\begin{align*}
	&|B_2|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a(\bar{\mathrm{v}}_{t,x_2}) a(\mathrm{u}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \overline{\omega_{N,t}(x_1;z_1) } \\
	&
	\qquad \int  \mathrm{d}\eta \mathrm{d}\eta'\ a_{\eta} a_{\eta'} \mathrm{v}_t(\eta;x_2) \overline{\mathrm{u}_t(\eta'; z_2)}   \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\    \big<  \xi_N, \mathcal{U}^*_N(t;0) , O_1(x_1;z_1) \omega_{N,t}(z_1;x_1)  \\
	&
	\qquad \int  \mathrm{d}\eta \mathrm{d}\eta'\ a_{\eta} a_{\eta'} \mathrm{v}_t(\eta;x_2) O_2(x_2;z_2) \mathrm{u}_t(z_2;\eta')  \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	= \bigg| \int  \mathrm{d}\eta \mathrm{d}\eta'  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left(\int \mathrm{d}x_1\  \big(O_1\omega_{N,t}\big)(x_1;x_1) \right) a_{\eta} a_{\eta'} \big(\mathrm{v}_t O_2 \mathrm{u}_t\big)(\eta;\eta')  \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \trnorm{O_1 \omega_{N,t}} \hsnorm{\mathrm{v}_t O_2 \mathrm{u}_t}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \hsnorm{\mathrm{v}_t} \opnorm{O_2}  \opnorm{\mathrm{u}_t}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq N  \hsnorm{O_1}  \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N},
\end{align*}
where we use the fact that
\[
\left| \Tr O_1 \omega_{N,t} \right|\leq \trnorm{O_1 \omega_{N,t}} \leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \leq \sqrt{N} \hsnorm{O_1}.
\]


\begin{align*}
	&|B_3|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a(\bar{\mathrm{v}}_{t,x_1}) a(\mathrm{u}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \mathrm{u}_t(z_1;x_2)\\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a_{\eta} a_{\eta'} \mathrm{v}_t(\eta;x_1) \overline{\mathrm{u}_t(\eta'; z_2)}  \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\int  \mathrm{d}\eta \mathrm{d}\eta'\big<  \xi_N, \mathcal{U}^*_N(t;0) , a_{\eta} a_{\eta'} \\
	&
	\qquad  \mathrm{v}_t(\eta;x_1) O_1(x_1;z_1 )\mathrm{u}_t(z_1;x_2) O_2(x_2;z_2) {\mathrm{u}_t(z_2;\eta')}  \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int  \mathrm{d}\eta \mathrm{d}\eta'\big<  \xi_N, \mathcal{U}^*_N(t;0) , a_{\eta} a_{\eta'}  \big(\mathrm{v}_t O_1 \mathrm{u}_t O_2 \mathrm{u}_t\big)(\eta;\eta')  \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t O_2 \mathrm{u}_t}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N} \\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_4|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a^*(\bar{\mathrm{v}}_{t,z_2})  a(\bar{\mathrm{v}}_{t,x_1})  \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{\mathrm{v}_t O_1 \mathrm{u}_t O_2 \bar{\mathrm{v}}_t} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N} \\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_5|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_2} \right> \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \mathrm{u}_t(z_1;x_2) \omega_{N,t}(z_2;x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg|  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \int \mathrm{d}x_1 \mathrm{d}x_2   \big(O_1 \mathrm{u}_t\big)(x_1;x_2)\big( O_2 \omega_{N,t}\big)(x_2;x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	= \bigg|  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \int \mathrm{d}x_1   \big(O_1 \mathrm{u}_t O_2 \omega_{N,t}\big)(x_1;x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg|\\
	&
	\leq \hsnorm{ O_1 \mathrm{u}_t O_2} \hsnorm{\omega_{N,t}} \\
	&
	\leq \sqrt{ N} \hsnorm{O_1} \opnorm{O_2} .
\end{align*}

\begin{align*}
	&|B_6|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_2}) a(\mathrm{u}_{t,z_2}) \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_1) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \mathrm{u}_t(\eta;x_2) \overline{\mathrm{u}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	= \bigg| \int  \mathrm{d}\eta \mathrm{d}\eta'\big<  \xi_N, \mathcal{U}^*_N(t;0) , \int \dd{x_1} \big(O_1\omega_{N,t}\big) (x_1;x_1) a^*_{\eta} a_{\eta'} \big(\mathrm{u}_t {O_2} \mathrm{u}_t\big)(\eta;\eta') \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \opnorm{\mathrm{u}_t {O_2} \mathrm{u}_t}  \norm{\xi_N} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}
\end{align*}


\begin{align*}
	&|B_7|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_2})  a^*(\bar{\mathrm{v}}_{t,z_2})  \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_1) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a^*_{\eta'} \mathrm{u}_t(\eta;x_2) \overline{\mathrm{v}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \hsnorm{\mathrm{u}_t O_2 \bar{\mathrm{v}}_t}  \norm{\xi} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \sqrt{N} \hsnorm{O_1} \opnorm{\mathrm{u}_t}\opnorm{O_2} \hsnorm{\mathrm{v}_t} \norm{\xi_N} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{8}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right>  a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \mathrm{u}_t(\eta;x_1) \overline{\mathrm{u}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \opnorm{\mathrm{u}_1 O_1 \omega_{N,t} O_2 \mathrm{u}_t} \norm{\xi} \norm{\mathcal{N} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \opnorm{O_1} \opnorm{O_2} \opnorm{ \omega_{N,t}}  \norm{\xi_N} \norm{\mathcal{N} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \norm{\xi} \norm{\mathcal{N} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{9}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1} \right> a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2})   \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a^*_{\eta'} \mathrm{u}_t(\eta;x_1) \overline{\mathrm{v}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \overline{\omega}_{N,t} O_2 \bar{\mathrm{v}}_t} \norm{\xi_N} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{10}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \left<\mathrm{u}_{t,z_1}, \mathrm{u}_{t,x_2} \right> a^*(\mathrm{u}_{t,x_1})a(\mathrm{u}_{t,z_2})  \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \mathrm{u}_t(z_1;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \mathrm{u}_t(\eta;x_1) \overline{\mathrm{u}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \mathrm{u}_t O_2 \mathrm{u}_t}\norm{\xi_N} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{11}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,  \left<\mathrm{u}_{t,z_1},\mathrm{u}_{t,x_2} \right> a^*(\mathrm{u}_{t,x_1})a^*(\bar{\mathrm{v}}_{t,z_2})  \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) , \mathrm{u}_t(z_1;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a^*_{\eta'} \mathrm{u}_t(\eta;x_1) \overline{\mathrm{v}_t(\eta'; z_2)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{\mathrm{u}_t O_1 \mathrm{u}_t O_2 \bar{\mathrm{v}}_t} \norm{\xi_N} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2} \norm{(\mathcal{N}+1)^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{12}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,   \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2}\right> a^*(\bar{\mathrm{v}}_{t,z_1}) a(\bar{\mathrm{v}}_{t,x_1}) \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_2;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \overline{\mathrm{v}_t(\eta;z_1)} \mathrm{v}_t(\eta'; x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \opnorm{O_2} \trnorm{\omega_{N,t}} \opnorm{\bar{\mathrm{v}}_t \overline{O^*_1}  \mathrm{v}_t}  \norm{\xi_N} \norm{\mathcal{N} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq N \hsnorm{O_1} \opnorm{O_2}   \norm{\mathcal{N}  \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{13}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,   \left<\bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_1}\right> a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_1}) \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \overline{\mathrm{v}_t(\eta;z_2)} \mathrm{v}_t(\eta'; x_1) \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq  \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_2} \overline{\omega}_{N,t} \overline{O^*_1}  \mathrm{v}_t}  \norm{\xi_N} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{14}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,   \left< \bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2} \right> a^*(\mathrm{u}_{t,x_1}) a(\mathrm{u}_{t,z_1})  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_2;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \mathrm{u}_t(\eta; x_1)\overline{\mathrm{u}_t(\eta)} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq  \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_2} \overline{\omega}_{N,t} \overline{O^*_1}  \mathrm{v}_t}  \norm{\xi_N} \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}\\
	&
	\leq \hsnorm{O_1} \opnorm{O_2}  \norm{\mathcal{N}^{1/2} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}

\begin{align*}
	&|B_{15}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,    \left< \bar{\mathrm{v}}_{t,x_2}, \bar{\mathrm{v}}_{t,z_2} \right> a^*(\bar{\mathrm{v}}_{t,z_1}) a^*(\mathrm{u}_{t,x_1}) \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_2;x_2) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a^*_{\eta'} \overline{\mathrm{v}_t(\eta;z_1)} \overline{\mathrm{u}_t(x_1;\eta')} \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \trnorm{O_2 \omega_{N,t}} \hsnorm{\bar{\mathrm{v}}_t \overline{O^*_1} \bar{\mathrm{u}}_t} \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \hsnorm{\mathrm{v}_t} \opnorm{O_2} \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq N \hsnorm{O_1}\opnorm{O_2} \norm{\mathcal{N}^\frac{1}{2} \mathcal{U}_N(t;0) \xi_N}.
\end{align*}

\begin{align*}
	&|B_{16}|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,   \left<\bar{\mathrm{v}}_{t,x_1}, \bar{\mathrm{v}}_{t,z_1}\right> a^*(\bar{\mathrm{v}}_{t,z_2}) a(\bar{\mathrm{v}}_{t,x_2})  \mathcal{U}_N(t;0)  \xi_N \big>   \bigg|\\
	&
	= \bigg| \int \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}z_1 \mathrm{d}z_2\ O_1(x_1;z_1) O_2(x_2;z_2)  \big<  \xi_N, \mathcal{U}^*_N(t;0) ,\omega_{N,t}(z_1;x_1) \\
	&
	\qquad  \int  \mathrm{d}\eta \mathrm{d}\eta'\ a^*_{\eta} a_{\eta'} \overline{\mathrm{v}_t(\eta;z_2)} \mathrm{v}_t(\eta'; x_2) \mathcal{U}_N(t;0)  \xi_N \big> \bigg| \\
	&
	\leq \hsnorm{O_1} \hsnorm{\omega_{N,t}} \opnorm{\bar{\mathrm{v}}_t \overline{O^*_2} \mathrm{v}_t} \norm{\xi_N} \norm{\mathcal{N} \mathcal{U}_N(t;0) \xi_N}\\
	&
	\leq  \sqrt{N} \hsnorm{O_1} \opnorm{O_2} \norm{ \mathcal{N} \mathcal{U}_N(t;0)  \xi_N}.
\end{align*}
\end{appendices}



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