\section{Introduction}
\label{intro}
\par One of the standard problems in classical information theory is the
following: given two probability distributions, find out a stochastic method so
that they can be distinguished. This can be answered probabilistically and the
relevant area of research is known as hypothesis testing. A generalization of this
problem with more than two probability distributions is known as multiple
hypothesis testing. The quantum version of these  problems, i.e. quantum
hypothesis testing also has a rich history. One can see the book of Hayashi
\cite{hayashi-1} and the references therein to know the history of the problem.

\par in this paper, we consider a quantum version of the problem from a matrix
analysis point of view. Following the notion of Kolmogorov, a probability space
can be represented by a triplet $(\Omega, \Sigma, f)$, where $\Omega$ is the
sample space, $\Sigma$ is the set of events, and $f$ is a probability
distribution. A quantum probability space can similarly be represented by a
triplet $(H, \mathcal{P}(H), \rho)$, where sample space is represented by a
complex separable Hilbert space $H$, set of events is represented by the set of
projections on $H$ denoted by $\mathcal{P}(H)$, and a probability distribution is
replaced by a positive semi-definite trace class Hermitian operator $\rho$ where
$tr[\rho]=1$. This is called a state. If $\rho$ is a rank one projection, it is
also called a pre-state. See the book of Meyer \cite{meyer} or Parthasarathy  \cite{krp6}
for more details regarding this approach. in this paper, we only consider Hilbert
spaces of finite dimensions. As a result, the operators involved are all matrices.
Physical transformations between two quantum systems are given by completely
positive trace-preserving transformations, which are called quantum channels.
A quantum channel $\Phi: \mathcal{B}(H) \to  \mathcal{B}(H)$ can be represented
by a finite set of operators $\{V_1,V_2, \cdots, V_k\}$ acting on $H$ such that
for all $X \in \mathcal{B}(H)$
\begin{equation}\label{eq1.1}
	\Phi(X) = \sum_{j=1}^k V_j X V_j^*,   \text{ where }
	\sum_{j=1}^k V_j^* V_j=I.
\end{equation}
Here $I$ denotes the identity operator on $H$. The operators $V_j$ are called the
Kraus operators. (See the chapter 3 from the book of  Bhatia \cite{bhatia1} for a  proof.
This representation is not unique.
However, a minimal representation with respect to the number of terms can be achieved. 


\par The problem can be stated as follows: Let $A_1,\cdots, A_n$ be a set of
input states and $B_1, \cdots, B_n$ be a set of output states, both on
$\mathcal{B}(H)$. The task is to construct (if possible) a quantum channel $\Phi:
\mathcal{B}(H) \to \mathcal{B}(H)$ such that $\Phi(A_i) =B_i$ for each $i=1,\, 2,
\cdots, n.$ This can be considered as a version of interpolation problem in
operator theory where an interpolating function is a completely positive trace
preserving map.  When $\dim H=2$ and $n=2$ the problem has been completely solved
by  Alberti and Uhlmann \cite{alberti-uhlmann-80}, with some more developments in
\cite{MR649773}. Further developments, in the sense of hypothesis testing, was done
by Jen\v{c}ov\'{a} \cite{MR2997404},\cite{MR2661520}. However, the most general form of the above problem remains elusive. The classical version of the problem was solved by Blackwell
\cite{MR56251}. For further developments in this direction see the paper of Ruch
et al \cite{MR586660} or the book of Torgersen \cite{MR1104437}. All of these
processes use various properties of matrix majorization. 


\par The statement of the problem can be explained in the language of operator
systems. Consider the operator system $\mathcal{S}_A$ spanned by $A_i$'s and the
map $\phi:\mathcal{S}_A \to \mathcal{B}(H)$ such that $\phi(A_i)=B_i$. By
definition, this is a completely positive map. By Arveson's extension theorem
this can be extended to a completely positive map $\Phi: \mathcal{B}(H) \to
\mathcal{B}(H)$ \cite{paulsen}. This extension is not unique and it is not clear
whether this extension is  a quantum channel or not. 

\par Li and Poon approached this problem from a matrix analysis point of
view \cite{MR2837768}.
They assumed that the input and output matrices are also commuting. As a result,
the problem is reduced to a problem on diagonal matrices, which can be approached by
standard methods.   This had been further
extended by Hsu et al \cite{hsu-jfa-14} where they considered only one input $A$
and only one output $B$ where the operators are either compact or of Schatten
$p$-class. Following techniques as in \cite{MR2837768}, they showed that the
result also holds for a class (compact or Schatten $p$) of input operators
$A_i$'s and corresponding output operators $B_i$'s where they are mutually
commuting.

\par From a quantum information perspective Heinosaari et al \cite{heinosaari-12}
applied semi-definite programming techniques to construct such channels. They also
asked a sequential version of the problem, i.e. to say whether there exists a
sequence of quantum channels $\Phi_j$ such that $\lim_{j\to \infty} \|\Phi_j(A_i)
-B_i\|=0$ for all $i=1, \cdots, n$. They observed that there exist cases when the sequential version of the problem has solutions whereas there does not exist
any such channel $\Phi$. Finally, Gour et al  \cite{Gour2018} showed several
equivalent statements of the problem connecting it with quantum thermodynamics.
Following the classical problem, which relies heavily on majorization, they
consider the problem as a non-commutative version of majorization. 

\par In \cite{MR3671511} Davidson et al approached interpolation problems from an
operator theoretic perspective. They did these works in the context of $d$-tuples
of operators on Complex Hilbert space and they used duality of the Hilbert space
most of their work. Basically, they associate with d-tuples of bounded operators
to the matrix range which is introduced by Arveson.
That is d-tuple of operators
can be mapped onto another d-tuple of operators by a UCP map. This result
connects the problem with the joint numerical range which was also initially noticed
in  \cite{hsu-jfa-14} in a very restricted class of operators. 


\par A quantum channel $\Phi$ is said to be
entanglement breaking if for any state $\rho$ acting on $H \otimes H$ the output
$(1 \otimes \Phi) (\rho)$ is a separable state. This has been introduced by
Horodecki et al \cite{MR2001114}. It can be shown that for such a map $\Phi$
there is a Kraus representation (which need not be optimal) such that each Kraus
operator is of rank 1. in this paper, we tried to address the following questions:
\begin{enumerate}
 \item First we have solved the interpolation problem with entanglement breaking
	 map when initial states are orthogonal.
 \item Secondly we have reconstructed the SDP's in \cite{heinosaari-12} so that
	 can identify the EBT maps.
 \end{enumerate}
 We notice that Li and Du \cite{MR3336734} also had addressed a similar problem.
 But in their case, there was one input and one output operator and the separable
 Hilbert spaces were of any arbitrary dimensions. in this paper, we consider the
 problem only on finite dimensional Hilbert spaces. 

 \par The paper has been organized as follows.
 \begin{enumerate}
     \item In section \S2 we have stated some preexisting results which have played important roles in our original discussion.
     \item In section \S3 we have addressed the first question.
     We have found out the necessary and sufficient conditions for the existence of an entangle breaking map $\phi$ such that $\phi(A_i)=B_i$ assuming that $A_i's$ are orthogonal with respect to the standard inner product of matrices.
     Further, we have determined a necessary and sufficient condition that the entanglement breaking map $\phi$ would be an entanglement breaking trace-preserving map(EBT).
     We have also given a method to explicitly construct $\phi$.
     \item section \S4 is divided into two subsections.
     
     In subsection \S4.1 we have given a sequential convex optimization problem(i.e the constraints of the last optimization is determined by the previous optimization and so on) to identify the interpolating EBT maps.
     
     In subsection \S4.2 we have given a dif and only iferent convex optimization for the same purpose.
     This time the optimization problem has some nice structure. This is a cone program and the only dif and only iference it has from an SDP is that its constraints are running over a smaller convex cone of semi-definite matrices rather than the full set.
     For this, we are denoting this optimization as a semi-SDP.
     \item In Section, \S5 we have taken a diversion from our previous discussions.
     Here we have treated random unitary channels in the same way we have treated EBT channels in Subsection \S4.2 and the resulting cone program had a striking similarity with the cone program in Subsection \S4.2.
     \item Finally, in section \S6, we have generalized our discussions in subsection \S4.2 and section \S5.
     We have shown here that the interpolation problem can be reformulated as a cone program whose constraints are running over a convex cone of semi-definite matrices(semi-SDP) as long as the interpolating quantum channels are coming from a convex set.
 
 \end{enumerate}

\section{Preliminary Concepts and Results}
\label{sec:1}
In this section, we are going to list out the preexisting results we are going to use in our discussion.\\

 Let $ \mathbb{M}_n$ be the set of all complex matrices and $ \mathbf{H}_n$ be the set of all Hermitian matrices in $ \mathbb{M}_n$. A matrix $A\in \mathbb{M}_n$ is said to be positive if   $ A \in \mathbf{H}_n$ and all of its eigenvalues are nonnegative. A linear map $ \Phi : \mathbb{M}_n \rightarrow  \mathbb{M}_m$ is positive if it maps positive matrices to positive matrices. Let, $\mathbb{M}_k(\mathbb{M}_m)$ be the space of all $k\times k$  block matrices $[[A_{ij}]]$ where $A_{ij} \in \mathbb{M}_m$. A linear map $ \Phi : \mathbb{M}_n \rightarrow  \mathbb{M}_m$ is completely positive if for each k, the map $  I  \otimes \Phi :\mathbb{M}_k(\mathbb{M}_n) \rightarrow \mathbb{M}_k(\mathbb{M}_m)$ defined by $ (I_k \otimes \Phi)(A_{ij})=[\Phi(A_{ij})] $ is positive.\\
 
 In the Introduction, we said interpolation problem as a version of operator theory.
 Now we want to see for a given set of input states $\{A_1,A_2,\dots,A_n\}$ and output states $\{B_1,B_2,\dots,B_n\}$,
 does there exists a condition for which existence of a completely positive trace-preserving maps $ \Phi : \mathbb{M}_n \rightarrow  \mathbb{M}_m$ such that $\Phi(A_i)=B_i$ for $i=1,2,\dots,n$?
 In particular, for $i=1$, that is for single matrix A and B, can we find a condition for the existence of completely positive trace-preserving maps such that $ \Phi(A)=B$?
 The following results by Li and Poon show that condition for the existence of a completely positive trace-preserving maps for Hermitian matrices.\\

\begin{theorem}\label{th2.1}\cite{MR2837768}
Suppose  $A\in\mathbf{H}_n$ and $B\in\mathbf{H}_m$ have eigenvalues $a_1,\dots,a_n$ and $b_1,\dots,b_m$ respectively.
Let $ a=(a_1,\dots,a_n)$ and $b=(b_1\dots ,b_m)$.
The following condition are equivalent:
\begin{enumerate}
 \item There is a completely positive map trace-preserving  $\Phi:\mathbb{M}_n\rightarrow\mathbb{M}_m$ such that $\Phi(A)=B$
 \item There exists an $n\times m$ row stochastic matrix $\mathbf{D}$ such that $b=a\mathbf{D}$.
\item We have $tr(A)=tr(B)$ and $\sum_{p=1}^n|a_p|\geq\sum_{q=1}^m|b_q|$.
 \end{enumerate}
 \end{theorem}
 
 After adding some condition on the matrices we can find the condition of the existence of completely positive trace-preserving map for finite input and output matrices. More precisely, If we consider inputs $\{A_1,A_2,\dots,A_n\}$ and outputs $\{B_1,B_2,\dots,B_n\}$ are commuting family of Hermitian matrices. Then the existence of a completely positive trace-preserving for above family of matrices such that $\Phi(A_i)=B_i$ for $i=1,2,\dots,n$ as follows.
 
 \begin{theorem}\label{th2.2}\cite{MR2837768}
Let $\{A_1,\dots ,A_k\}$ in $\mathbf{H}_n$ and $\{B_1,\dots , B_k\}$ in $\mathbf{H}_m$ be two commuting families.
Then there exist  unitary matrices $U\in\mathbb{M}_n ~and~V\in\mathbb{M}_m$ such that $U^*A_iU$ and $VB_iV^*$ are diagonal matrix with diagonals $a_i=(a_{i1},\dots ,a_{in})$ and $b_i=(b_{i1},\dots ,b_{im})$ respectively, for i=1,...k. The following conditions are equivalent:

\begin{enumerate}
\item There is a completely positive map $\Phi : \mathbb{M}_n \rightarrow \mathbb{M}_m$ such that $\Phi  (A_i)=B_i $ for $i=1,\dots ,k$.
\item There is an $n\times m$ non-negative matrix $D=\begin{pmatrix}
d_{pq}
\end{pmatrix}_{pq}$ such that $(b_{ij})=(a_{ij})D$.
\end{enumerate}

\end{theorem}
 

 Now from  the definition of entanglement breaking we know that any output state of an entanglement breaking channel is separable.
 From \cite{H1}, we also know how to check the separability of a state.
 The following result from \cite{MR2001114} shows that how to characterize entanglement breaking maps. 

\begin{theorem}\label{th2.3}
Let $\phi$ be a completely positive map on $\mathcal{B}(H_1)\otimes\mathcal{B}(H_2)$ where $H_1=H_2=\mathbb{C}^n$.
Then the following are equivalent:
\begin{enumerate}
    \item $\phi$ can be written as $\phi(\rho)=\sum_k R_ktr[F_k\rho]$\footnote{This form of $\phi$ is called Holevo form, see \cite{Holevo-coding}} with $F_k$ positive semi-definite for every $k$.
    \item $\phi$ is entanglement breaking.
    \item $ I \otimes\phi(|\beta\rangle\langle\beta|)$ is separable for any maximally entangled state $\beta=n^{-\frac{1}{2}}\sum_{j=1}^n |j\rangle\otimes|j\rangle$.
    \item $\phi$ can be written in operator sum form using only Kraus operators $\{A_l\}_l$ of rank one.
    \item $\gamma\circ\phi$ is completely positive for all positivity preserving maps $\gamma$.
    \item $\phi\circ\gamma$ is completely positive for all positivity preserving maps $\gamma$.
\end{enumerate}
A corresponding equivalence holds for CPT and EBT maps with the additional
conditions that $\sum_k F_k= I $, the Kraus operators $A_l$ satisfy $\sum_l A_lA_l^*= I $,
and $\gamma$ is trace-preserving.
\end{theorem}


Using these theorems we have found a completely positive trace-preserving map between two families of positive matrices where initial matrices are orthogonal and we have also found whether the map is entanglement breaking or not.

The second problem is inspired by the existence of a completely positive trace-preserving extension formulated as a semi-definite program in \cite{heinosaari-12}.\\

Let $\{X_i\}_{i=1}^N$ and $\{Y_i\}_{i=1}^N$ be Hermitian matrices.
If linear map $\tilde{T}:\mathbb{M}_d\rightarrow \mathbb{M}_{d'}$ is the approximation of the linear map $T:\mathbb{M}_d\rightarrow \mathbb{M}_{d'}$ such that $T(X_i)=Y_i$. Define this $\Delta$ to be a measure of the  functional\\
 \begin{equation}\label{eq2.1}
\Delta(\tilde{T})=\sum_{i=1}^N\Vert\tilde{T}(X_i)-Y_i\Vert_1 
\end{equation}
which calculate a linear map $\tilde{T}$ is how much close to the linear map $ T$.\\

So,
\begin{equation}\label{eq2.2}
\Delta=\inf\{\Delta(\tilde{T})|\tilde{T}:\mathbb{M}_d\rightarrow\mathbb{M}_{d'}~ ~is~completely~~positive\} 
\end{equation}
calculates how well the linear map $T:\mathbb{M}_d\rightarrow \mathbb{M}_{d'}$ such that $T(X_i)=Y_i$ can be approximated by a completely positive map defined on $\mathbb{M}_d$.
 Heinosaari et al \cite{heinosaari-12} have identified all this as the following SDP problem.
 
\begin{theorem}\label{th2.4}
Let, $X_i\in\mathbb{M}_d$ and $Y_i\in\mathbb{M}_{d'}$ be Hermitian matrices.
The optimization \ref{eq2.1}-\ref{eq2.2} whose optimal value $\Delta$ quantifies how well the map $X_i\rightarrow Y_i$  can be approximated with a completely positive map on $\mathbb{M}_d$ can be formulated as an SDP.
Up to negative sign, it is the dual of the following SDP\\
\begin{equation}\label{2.3}
\Gamma=\inf\sum_{i=1}^Ntr[Y_i^TH_i]
\end{equation}
Subject to constraints,
\begin{equation}\label{2.4}
\sum_{i=1}^NX_i\otimes H_i\geq0\\
~~and ~~\Vert H_i \Vert \leq 0 ~~\forall i
\end{equation}
where $H_i$'s are Hermitian matrices.
The optimal value $\Gamma$ of the SDP \ref{2.3} - \ref{2.4} is equal to $0$ if and only if the map $X_i\rightarrow Y_i $ is completely positive on $\mathcal{S}=span\{X_i\}$; The optimal values both of the SDP are related by $ \Delta = -\Gamma$. 
\end{theorem}
In the paper, \cite{heinosaari-12} the existence of a completely positive trace-preserving extension formulate as the following two semi-definite program.\\

The first one is:

\begin{equation}\label{eq2.5}
\Delta_1=\inf\{\omega\lambda+\sum_i tr[P_i+Q_i]\} 
\end{equation}
Subject to
\begin{equation}\label{eq2.6}
C,P_i,Q_i\geq0 
\end{equation}
\begin{equation}\label{eq2.7}
P_i-Q_i=tr_2[C( I \otimes X_i^T)]-Y_i~~\forall i 
\end{equation}
\begin{equation}\label{eq2.8}
-\lambda I \leq tr_1[C]- I \leq\lambda I 
\end{equation}

The second one is:
\begin{equation}\label{eq2.9}
\Delta_2=\inf\{\sum_i tr[P_i+Q_i]\} 
\end{equation}
Subject to
\begin{equation}\label{eq2.10}
C,P_i,Q_i\geq 0 
\end{equation}
\begin{equation}\label{eq2.11}
P_i-Q_i = tr_2[C( I \otimes X_i^T)]-Y_i~~\forall i 
\end{equation}
\begin{equation}\label{eq2.12}
tr_1[C] =  I 
\end{equation}
 
We have modified these SDP's in the subsequent sections to identify EBT maps.








\section{An Explicit construction of Interpolating Entanglement Breaking Map When The Given Initial States Are Orthogonal}

In this section, we are going to solve the interpolation problem with entanglement breaking map for a very restricted case, i.e when the set of initial states $\{ A_i\}_{i=1}^k$ is orthogonal.\\

Moreover, we are going to prove that before mentioned entanglement breaking map is an EBT if $ I \in span(\{ A_i\}_{i=1}^k)$.\\

Our main result for this section is:
\begin{theorem}\label{th3.1}
Let $\mathcal{A}=\{ A_1,A_2,...,A_k\}$ be an orthogonal set of positive matrices and $\mathcal{B}=\{ B_1,B_2,...,B_k\}$ be another set of positive matrices. Then the following statements are equivalent:

(a) $tr[A_i]=tr[B_i]$ for each $i\in\{1,2,...,k\}$.

(b) There is an entanglement breaking map $\phi:\mathbb{M}_n\rightarrow\mathbb{M}_n$ such that $\phi(A_i)=B_i$ for each $i\in\{1,2,...,k\}$ and $\phi$ is trace-preserving on $span(\mathcal{A})$.
\end{theorem}

To prove this theorem we have to prove a few lemmas first.
Those are:
\begin{lemma}\label{lm3.2}
Let $V$ be a one dimensional subspace of $\mathbb{M}_n$ spanned by a positive matrix $A$ with $\langle A,A\rangle=1$.
Let $P_V$ be the projection map onto $V$ and let $\phi$ be a completely positive map on $\mathbb{M}_n$.
Then $\phi_V=\phi\circ P_V$ is completely positive.
\end{lemma}

\begin{proof}
By definition of projection map, $P_V(B)=\langle B,A\rangle A=tr[B^*A]A$ for any $B\in\mathbb{M}_n$.
Let assume, $A=\begin{pmatrix}
a_{ij}
\end{pmatrix}_{ij}$

Then, $P_V(E_{rs})=tr[E_{rs}^*A]A=tr[E_{sr}A]A=a_{sr}A$.

Then $ I \otimes\phi_{V}\begin{pmatrix}
(\begin{pmatrix}
E_{ij}
\end{pmatrix}
\end{pmatrix}_{ij}=\begin{pmatrix}
\begin{pmatrix}
\phi_V(E_{ij})
\end{pmatrix}
\end{pmatrix}_{ij}=\begin{pmatrix}
\begin{pmatrix}
\phi(a_{ji}A)
\end{pmatrix}
\end{pmatrix}_{ij}=\begin{pmatrix}
\begin{pmatrix}
a_{ji}\phi(A)
\end{pmatrix}
\end{pmatrix}_{ij}=\begin{pmatrix}
a_{ji}
\end{pmatrix}_{ij}\otimes\phi(A)=A^T\otimes\phi(A)$ is positive.

Hence $\phi_V$ is completely positive.
\end{proof}

\begin{lemma}\label{lm3.3}
Let $\mathcal{A}=\{ A_1,A_2,...,A_k\}$ be an orthogonal set of positive matrices and $\mathcal{B}=\{ B_1,B_2,...,B_k\}$ be another set of positive matrices. Then the following statements are equivalent:

(a) $tr[A_i]=tr[B_i]$ for each $i\in\{1,2,...,k\}$.

(b) There is a completely positive map $\phi:\mathbb{M}_n\rightarrow\mathbb{M}_n$ such that $\phi(A_i)=B_i$ for each $i\in\{1,2,...,k\}$ and $\phi$ is trace-preserving on $span(\mathcal{A})$.
\end{lemma}
\begin{proof}
Let us assume (a).
By theorem \ref{th2.1}, there are CPT maps $\phi_i:\mathbb{M}_n\rightarrow\mathbb{M}_n$ such that $\phi_i(A_i)=B_i$ for each $i\in\{1,2,...,k\}$.

Let us define $\Tilde{\phi}_i=\phi_i\circ P_{A_i}$ where $P_{A_i}$ is the projection map onto the subspace spanned by $A_i$.
Then by lemma \ref{lm3.2}, $\Tilde{\phi}_i$ is completely positive for every $i$.

Let define $\phi=\sum_{i=1}^k\Tilde{\phi}_i$.

Then, $\phi(A_j)=\sum_{i=1}^k\Tilde{\phi}_i(A_j)=\sum_{i=1}^k\phi_i(P_{A_i}(A_j))=\sum_{i=1}^k\phi_i(\frac{\langle A_j,A_i\rangle}{\langle A_i,A_i\rangle}A_i)=\sum_{i=1}^k\phi_i(\delta_{ij}A_i)=\sum_{i=1}^k\delta_{ij}\phi_i(A_i)=\phi_j(A_j)=B_j$.

Moreover, $ I \otimes\phi\begin{pmatrix}
\begin{pmatrix}
E_{rs}
\end{pmatrix}
\end{pmatrix}=\sum_{i=1}^k  I \otimes\Tilde{\phi}_i\begin{pmatrix}
\begin{pmatrix}
E_{rs}
\end{pmatrix}
\end{pmatrix}=\sum_{i=1}^k A_i^T\otimes\phi_i(A_i)=\sum_{i=1}^k A_i^T\otimes B_i$.
Hence clearly $ I \otimes\phi\begin{pmatrix}
\begin{pmatrix}
E_{rs}
\end{pmatrix}
\end{pmatrix}$ is positive and therefore $\phi$ is completely positive.

Let $A\in span(\mathcal{A})$ be arbitrary.
Then $A=\sum_{i=k}^k\alpha_iA_i$ for some $\alpha_i\in\mathbb{C}$.

Hence, $\phi(A)=\sum_{i=k}^k\alpha_i\phi(A_i)=\sum_{i=k}^k\alpha_iB_i$.

Therefore, $Tr[\phi(A)]=Tr[\sum_{i=k}^k\alpha_iB_i]=\sum_{i=k}^k\alpha_iTr[B_i]=\sum_{i=k}^k\alpha_iTr[A_i]=Tr[\sum_{i=k}^k\alpha_iA_i]=Tr(A)$.

So $\phi$ is a completely positive map sending $A_i$ to $B_i$ and trace-preserving on $span(\mathcal{A}).$

Now (a) is obviously implied if we assume (b).

\end{proof}

Now to prove theorem \ref{th3.1}, we only need to prove that the $\phi$ we constructed in lemma \ref{lm3.3} is entanglement breaking.
By theorem \ref{th2.3} we know that one of the equivalent conditions of a map being entanglement breaking is that it has a Kraus operator representation such that all the Kraus operators are of rank one.
The proof is as follows:

\begin{proof}
It is enough to prove that the Kraus operators of $\Tilde{\phi}_i$ can be chosen to be of rank 1 for each $i$.
Let us fix $i$.

Let assume, $U^*A_iU$ and $VB_iV^*$ are diagonal matrices with diagonals $a_i=(a_{i1},a_{i2},...,a_{in})$ and $b_i=(b_{i1},b_{i2},...,b_{in})$ respectively and $U$ and $V$ are unitary matrices.
Since $\phi_i$ is a completely positive map taking $A_i$ to $B_i$, by theorem \ref{th2.2} there is an $n\times n$ non-negative matrix $D=\begin{pmatrix}
\begin{pmatrix}
d_{pq}
\end{pmatrix}
\end{pmatrix}$ such that $b_i=a_iD$.

Let define, $F_j=u_j^Te_j$ where $u_j=(\sqrt{d_{1j}},\sqrt{d_{2j}},...,\sqrt{d_{nj}})$ and $\{ e_i\}_{1=1}^n$ is the standard basis of $\mathbb{C}_n$.

Then $\Tilde{\phi}_i$ can be written as: $$\Tilde{\phi}_i(A)=\sum_{j=1}^n(UF_jV)^*A(UF_jV)$$

Clearly rank of $F_j$ is at most one for each $J$.
Therefore rank of $UF_jV$ is also at most one for each $j$.

Since $\Tilde{\phi}_j$ is a non-zero map rank of $UF_jV=1$ for at least one $j$.

Therefore $\Tilde{\phi}_i$ has a Kraus operator representation with only rank 1 Kraus operators.
\end{proof}

But theorem \ref{th3.1} has a big problem, which is it gives us a way to construct an entanglement breaking map interpolating between two given sets of positive matrices where the initial set is orthogonal, but the map is not necessarily trace-preserving and therefore not necessarily a quantum channel.
In the next corollary, we shall prove that the entanglement breaking map will be an EBT if $ I \in span(\mathcal{A})$.
\begin{corollary}
Let $\mathcal{A},\mathcal{B}$ and $\phi$ be as in theorem \ref{th3.1} with the extra property $ I \in span(\mathcal{A})$.
Then $phi$ is an EBT.
\end{corollary}

\begin{proof}
Let $\mathbb{A}=span(\mathcal{A})$ and $\mathbb{A}^\perp$ be the perpendicular space of $\mathbb{A}$.
Then any $V\in\mathbb{M}_n$ can be written as $V=V_1+V_2$ where $V_1\in\mathbb{A}$ and $V_2\in\mathbb{A}^\perp$.

Hence $Tr[\phi(V)]=Tr[\phi(V_1+V_2)]=Tr[\phi(V_1)]+Tr[\phi(V_2)]=Tr[V_1]+Tr[0]=Tr[V_1]$

Hence it is enough to prove that $Tr[W]=0$ for any $W\in\mathbb{A}^\perp$.

$ I \in\mathbb{A}\Rightarrow span( I )\subseteq\mathbb{A}\Rightarrow\mathbb{A}^\perp\subseteq span( I )^\perp$.

Let $P_ I $ be the projection map onto $span( I )$.

Then $Tr[P_ I (V)]=Tr[\frac{\langle I ,V\rangle}{\langle I , I \rangle} I ]=\frac{\langle I ,V\rangle}{\langle I , I \rangle}Tr[ I ]=Tr[V]$.

This implies that for any matrix $W\in span( I )^\perp$, $Tr[W]=0$.
\end{proof}

If we choose $\mathcal{A}$ and $\mathcal{B}$ to be sets of states then $Tr[A_i]=Tr[B_j]=1$ for any $i,j$.
Hence as long as we have an orthogonal set $\mathcal{A}$ of states such that $ I \in\mathbb{A}$ we can construct an EBT $\phi$ that takes $\mathcal{A}$ to $\mathcal{B}$ for any given set of states $\mathcal{B}$.

In this section, we have found out a very strong result about the existence of interpolating EBT maps between two given sets of states, but it imposes a lot of restrictions on the set of initial states.
In the subsequent sections we have tried to solve the same problem for the general case and reformulated those problems as some nice cone programs.

\section{An Optimization for Identifying EBT Maps}
\subsection{A Finite Sequence of Abstract Convex Optimizations}

In this section, our goal is to reconstruct the SDP's \ref{eq2.5}-\ref{eq2.8} and \ref{eq2.9}-\ref{eq2.12} so that we can identify the EBT maps.\\

We know from theorem \ref{th2.3} a map $T$ is EBT if and only if its corresponding Choi matrix $C$ is separable.
Using theorem 1, of \cite{H1} we can identify all separable states by an abstract convex optimization problem:

\begin{equation}\label{eq4.1}
\Lambda_C=\inf\{ tr[AC]\}
\end{equation}
 subject to
\begin{equation}\label{eq4.2}
tr[AR\otimes S]\geq 0 
\end{equation}
  for all projection matrices $R$ and $S$.\\

Optimization \ref{eq4.1}-\ref{eq4.2} is a convex optimization because the objective function $f_0(A)=tr[AC]$ and the constraint functions $f_{R,S}(A)=tr[AR\otimes S]$ are linear functions over a convex set.\\

Then again the constraint \ref{eq4.2} in the optimization \ref{eq4.1}-\ref{eq4.2} can be reformulated as the following abstract convex optimization problem:

\begin{equation}\label{eq4.3}
\Gamma_A=\inf\{ tr[AR\otimes S]\}
\end{equation}
subject to
\begin{equation}\label{4.4}
R^2 = R
\end{equation}

\begin{equation}\label{eq4.5}
S^2 = S
\end{equation}\\
Hence the optimization can be summarised as the following:\\

First, optimize $\Lambda_C$ over all $A$ for which $\Gamma_A\geq0$.
Then optimize $\Delta_i$ over all $C$ for which $\Lambda_C\geq0$.

\subsection{a cone program similar to an SDP}
In this section, we are going to use the definition of separability to construct a cone program that is almost an SDP(i.e just like an SDP, for this optimization the constraint functions run over a smaller convex cone of positive semi-definite matrices) that will identify the EBT maps interpolating between Hermitian $\{ X_i\}_i$ and $\{ Y_i\}_i$.\\

Before that, we have to note that in the SDP's \ref{eq2.5}-\ref{eq2.8} and \ref{eq2.9}-\ref{eq2.12}, $C$ is running over the convex cone of all positive semi-definite matrices.
On the other hand, this $C$ turns out to be Choi matrix of some completely positive map.
We know that for a quantum channel $\phi$, the corresponding Choi matrix $C_\phi$ must be a state in $\mathbb{M}_{n^2}$.
In spite of this fact, in the SDP's \ref{eq2.5}-\ref{eq2.8} and \ref{eq2.9}-\ref{eq2.12} $C$ were taken over all positive semi-definite matrices.\\

This can only mean that, if $\Delta_i=0$ then the minimum value of $w\lambda+\sum_{i}tr[P_i+Q_i]$ must be obtained for some $C$ with trace $1$.\\

\begin{lemma}\label{lm4.1}
This property will hold if we define an optimization similar to \ref{eq2.5}-\ref{eq2.8} where $C$ is in a subset $\mathcal{S}$ of positive semi-definite matrices instead of the full set and the same holds for \ref{eq2.9}-\ref{eq2.12}.
\end{lemma}
\begin{proof}
The proof follows from the fact that the minimum of a function over a subset is always greater than or equal to the minimum over the full set.\\

Let $\Delta_S=inf\{ w\lambda+\sum_i tr[P_i+Q_i]\}$ subject to $C\in\mathcal{S}$.
Then $\Delta_1\leq\Delta_S$.\\

Now, let us assume the claim in the lemma is not true. Then there is some $C\in\mathcal{S}$ with $tr[C]\neq 1$ and $\Delta_S=0$ for $C$.\\

That means $0\leq\Delta_1\leq\Delta_S=0\Rightarrow\Delta_1=0$ for $C$.
Hence the optimization identifies a completely positive map which is not a quantum channel.
This is a contradiction.\\

The proof for SDP \ref{eq2.9}-\ref{eq2.12} is similar.
\end{proof}

Keeping this property in mind let us define $\mathcal{C}=\{\sum_{i=1}^n p_iA_i\otimes B_i:p_i\geq 0,A_i$ and $B_i$ are states in $\mathbb{M}_n\forall i,\}$.
\begin{theorem}\label{th4.2}
$\mathcal{C}$ is a convex cone of positive semi-definite matrices.
\end{theorem}

\begin{proof}
Let choose $C_1,C_2\in\mathcal{C}$ and $\lambda\in(0,1)$ arbitrarily.
Let, $C_j=\sum_{i=1}^n p_{ji}A_{ji}\otimes B_{ji}$ for $i=1,2$.[Even if $C_1$ and $C_2$ have different number of components, we can adjust this by adding sufficiently many zeroes to the element with fewer components.]\\

Then, $\lambda C_1+(1-\lambda)C_2=\sum_{i=1}^n\lambda p_{1i}A_{1i}\otimes B_{1i}+\sum_{k=1}^n(1-\lambda)p_{2k}A_{2k}\otimes B_{2k}\in\mathcal{C}$.\\

Hence $\mathcal{C}$ is convex.\\

Let $r\in\mathbb{R}^+$ be chosen arbitrarily.
Clearly $rC\in\mathcal{C}$ for any $C\in\mathcal{C}$.\\

Hence $\mathcal{C}$ is a cone.
\end{proof}

Now we are in a position to construct our cone program.
We define the Optimization $\mathcal{O}$ as the following:

\begin{equation}\label{eq4.6}
\Delta =\inf\{ w\lambda+\sum_i tr[P_i+Q_i]\},
\end{equation}

subject to,
\begin{equation}\label{eq4.7}
P_i,Q_i\geq 0,
\end{equation}
\begin{equation}\label{eq4.8}
P_i-Q_i =tr_2[C( I \otimes X_i^T)]-Y_i\forall i,
\end{equation}
\begin{equation}\label{eq4.9}
C\in\mathcal{C},
\end{equation}
\begin{equation}\label{eq4.10}
    -\lambda I \leq tr_1[C]- I \leq\lambda I 
\end{equation}\\

There exists an interpolating EBT if and only if $\Delta=0$.

\section{An optimization problem to identify random unitary channels}
In this section, we are going to construct another optimization problem similar to \ref{eq4.7}-\ref{eq4.10} that will identify random unitary channels instead of EBT channels.\\

Our first step will be to characterize the Choi matrix of a random unitary channel.
Let $T$ be a random unitary channel.
Then $T(A)=\sum_{i=1}^n p_iU_i^*AU_i$ where $U_i$ is unitary and $p_i>0$ for each $i$ and $\sum_{i=1}^n p_i=1$.\\

Then the Choi matrix of $T$ is:
\begin{equation}\label{eq5.1}
    \begin{split}
        C_T =\begin{pmatrix}
\begin{pmatrix}
T(E_{ij})
\end{pmatrix}
\end{pmatrix}_{ij}\\
=\begin{pmatrix}
\begin{pmatrix}
\sum_{k=1}^np_kU^*_kE_{ij}U_k
\end{pmatrix}
\end{pmatrix}_{ij}\\
=\sum_{k=1}^np_k\begin{pmatrix}
\begin{pmatrix}
U^*_kE_{ij}U_k
\end{pmatrix}
\end{pmatrix}_{ij}
    \end{split}
\end{equation}\\

Hence from equation \ref{eq5.1}, we can conclude that the set of Choi matrices of random unitary channels are $\mathbb{C}_U=\{\sum_{k=1}^np_k\begin{pmatrix}
\begin{pmatrix}
U^*_kE_{ij}U_k
\end{pmatrix}
\end{pmatrix}_{ij}:U_k$ is unitary matrix in $\mathbb{M}_n$, $p_k>0$ for all $k$ and $\sum_{k=1}^np_k=1\} $.\\

As explained in section \S4.2, if the $\Delta_1$ in optimization \ref{eq2.5}-\ref{eq2.8} is zero, then the minimum value must have been obtained for some state $C$ and this is also true when $C$ is taken from a subset of positive semi-definite matrices instead of the full set.\\

Hence let us define, $\mathcal{C}_U=\{\sum_{k=1}^np_k\begin{pmatrix}
\begin{pmatrix}
U^*_kE_{ij}U_k
\end{pmatrix}
\end{pmatrix}_{ij}:U_k$ is unitary matrix in $\mathbb{M}_n$, $p_k>0$ for all $k\}$\\

Now the construction of the optimization problem will follow similar steps as in section \S3.2:

\begin{equation}\label{eq5.2}
\Delta  =\inf\{ w\lambda+\sum_i tr[P_i+Q_i]\},
\end{equation}

subject to,
\begin{equation}\label{eq5.3}
P_i,Q_i\geq 0,
\end{equation}
\begin{equation}\label{eq5.4}
    P_i-Q_i =tr_2[C( I \otimes X_i^T)]-Y_i\forall i,
\end{equation}
\begin{equation}\label{eq5.5}
    C\in\mathcal{C}_U
\end{equation}
\begin{equation}\label{eq5.6}
    -\lambda I \leq tr_1[C]- I \leq\lambda I 
\end{equation}

\section{A generalization of The Previous Discussions}
In our previous discussions, we have reconstructed the problem of interpolating EBT channels between two given sets of positive matrices as convex optimization problems.
In subsection \S4.2 we were even able to construct a cone program that is almost an SDP in the sense that the constraint functions run over a smaller convex cone of positive matrices rather than over the entire set of positive matrices.
In section \S5 we did a similar treatment for random unitary channels.
Let us denote this type of optimizations as semi-SDP.

\begin{definition}
We say a cone program $\mathcal{K}$ is a semi-SDP if its constraint functions run over a convex cone consisting of positive matrices.
\end{definition}

Now we are going to see how this methodology can be used in much more general case.
By general case what we mean is that instead of EBT or unitary channels we are going to impose an arbitrary property $\mathcal{P}$ on the set of quantum channels on $\mathbb{M}_n$ and try to check if there exist a quantum channel satisfying $\mathcal{P}$ and interpolating between two given sets of positive matrices.\\

Let us denote, $\mathbb{P}=\{ T:T$ is a quantum channel satisfying $\mathcal{P}\}$ as the corresponding set of $\mathcal{P}$.
We say $\mathcal{P}$ is convex if and only if $\mathbb{P}$ is convex.
We are going to show that for any convex $\mathcal{P}$ the before-mentioned problem can be represented as a semi-SDP.
By the one-one correspondence between the set of all completely positive maps and their corresponding Choi matrices $\mathbb{P}$ is convex if and only if the set of its corresponding Choi matrices is convex.
Therefore in our further discussion, we shall use the same notation $\mathbb{P}$
to denote the set of all quantum channels satisfying $\mathcal{P}$ as well as the set of Choi matrices corresponding to the former set.
\begin{definition}
Let $\mathbb{P}$ be the corresponding set of some property $\mathcal{P}$.
We define, $\mathbf{K}(\mathbb{P})=\{\sum_{i=1}^n p_iP_i: p_i\geq0$ and $P_i\in\mathbb{P}\forall i\}$.
\end{definition}

\begin{theorem}\label{th6.2}
For any $\mathbb{P}$, $\mathbf{K}(\mathbb{P})$ is a convex cone.
\end{theorem}

\begin{proof}
The proof is obvious.
\end{proof}

From this theorem we can say that $\mathbf{K}(\mathbb{P})$ is the convex cone generated by $\mathbb{P}$.\\

Here we have to note that, if we formulate a semi-SDP similar to \ref{eq4.6}-\ref{eq4.10} by running $C$ over $\mathbf{K}(\mathbb{P})$ it will not be necessarily true that the optimization may take $0$ value only if $C\in\mathbb{P}$.
If this happens then this cone program fails to identify interpolating quantum channels in $\mathbb{P}$.
But we shall see in the next theorem that if we take $\mathbb{P}$ to be convex then $\Delta=0$ only if $C\in\mathbb{P}$.\\

\begin{theorem}
Any arbitrary state in $\mathbf{K}(\mathbb{P})$ is in $\mathbb{P}$ if and only if $\mathbb{P}$ is convex.
\end{theorem}

\begin{proof}
First, we shall prove the set of all states in $\mathbf{K}(\mathbb{P})$ is $\mathbf{S}(\mathbb{P})=\{\sum_{i=1}^n p_iP_i: p_i\geq0,\sum_{i=1}^np_i=1$ and $P_i\in\mathbb{P}\forall i\}$.\\

Let $C=\sum_{i=1}^n p_iP_i\in\mathbf{S}(\mathbb{P})$.
Then $tr[C]=tr[\sum_{i=1}^n p_iP_i]=\sum_{i=1}^n p_itr[P_i]=\sum_{i=1}^n p_i=1$.\\

Hence $C$ is a state in $\mathbf{K}(\mathbb{P})$.\\

Let $D=\sum_{i=1}^n p_iP_i$ be an arbitrary state in  $\mathbf{K}(\mathbb{P})$.
Then $p_i\geq 0\forall i$ and $tr[D]=1$.\\

$1=tr[D]=tr[\sum_{i=1}^n p_iP_i]=\sum_{i=1}^n p_itr[P_i]=\sum_{i=1}^n p_i$.\\

Hence, $D\in\mathbf{S}(\mathbb{P})$ and therefore our claim at the beginning of the proof is true.\\

Now $\mathbf{S}(\mathbb{P})$ is the set of all convex combinations of the elements in $\mathbb{P}$.\\

We know that $\mathbf{S}(\mathbb{P})=\mathbb{P}$ if and only if $\mathbb{P}$ is convex.
Combining this with our previous claim we can conclude that any arbitrary state in $\mathbf{K}(\mathbb{P})$ is in $\mathbb{P}$ if and only if $\mathbb{P}$ is convex.
\end{proof}

We shall conclude our discussion of this section with the following theorem:

\begin{theorem}\label{th6.3}
Let $\mathcal{P}$ be a property on quantum channels and let $\mathbb{P}$ the set of quantum channels that satisfy $\mathcal{P}$.
If $\mathbb{P}$ is convex, then the problem of interpolation by elements in $\mathbb{P}$ can be reconstructed as the following semi-SDP:

\begin{equation}\label{eq6.1}
\Delta =\inf\{ w\lambda+\sum_i tr[P_i+Q_i]\}
\end{equation}

subject to,
\begin{equation}\label{eq6.2}
P_i,Q_i\geq 0
\end{equation}
\begin{equation}\label{eq6.3}
    P_i-Q_i =tr_2[C( I \otimes X_i^T)]-Y_i\forall i
\end{equation}
\begin{equation}\label{eq6.4}
    C\in\mathbf{K}(\mathbb{P})
\end{equation}
\begin{equation}\label{eq6.5}
    -\lambda I \leq tr_1[C]- I \leq\lambda  I 
\end{equation}

\end{theorem}

\section{conclusion}
In this paper we have first solved the famous interpolation problem of Quantum Information Theory for a very restricted class of operators where interpolating quantum channels were EBT.
After that, we turned our focus into reformulating the problem in discussion as some standard convex optimization problem, since there already exist algorithms to solve standard convex optimization problems and some of them can even be dealt with by computer programming.
In the process, we have constructed different convex optimization problems.
The first one is a cone program which is so similar to an SDP that the only difference between them is that its constraints are running over a smaller convex cone of positive semi-definite matrices rather than the full set. Then we did a similar treatment for interpolation problem by random unitary channels. At last we have drawn motivation for a generalized theory to show that for every property that forms a convex set of quantum channels our method can be implemented to reconstruct the corresponding interpolation problem as an conic program. 


\section*{Acknowledgement}
We are truly grateful to \textbf{Department of Science and Technology, Government of India} for providing Arnab Roy with \textbf{INSPIRE Fellowship} and \textbf{Council of Scientific and Industrial Research, Government of India} for providing Saikat Patra with \textbf{Junior Research Fellowship}.
The financial support we have received from these fellowships were crucial for our research.
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