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I am given fixed quantum states $\rho_X$ and $\sigma_Y$ and some function of the form $\text{Tr}(N_{X\rightarrow Y}(\rho_X)\sigma_Y)$. I would like to maximize this function over all completely positive trace preserving maps $N_{X\rightarrow Y}$. What is the way to solve such an optimization?

I notice that my problem is linear in $N_{X\rightarrow Y}$ so I was wondering if there could even be an analytical solution to this?

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For the specific linear function you are interested in, the solution turns out to be trivial: you can take the channel to be $N_{X\rightarrow Y}(\rho) = \operatorname{Tr}(\rho) |\psi\rangle\langle \psi|$ for $|\psi\rangle$ being an eigenvector of $\sigma_Y$ having the largest possible eigenvalue.

More generally, however, you can optimize any real-valued linear function over all channels of a fixed size using semidefinite programming. Perhaps the simplest way to do this is to use the Choi representation of the channels you are optimizing over: $$ J(N_{X\rightarrow Y}) = \sum_{a,b} N_{X\rightarrow Y}(|a\rangle\langle b|) \otimes |a\rangle\langle b|, $$ where $a$ and $b$ range over the standard basis states of $X$. It is the case that $N_{X\rightarrow Y}$ is a channel (i.e., is completely positive and trace preserving) if and only if $J(N_{X\rightarrow Y})$ is positive semidefinite and obeys the linear constraint $$ \operatorname{Tr}_{Y}(J(N_{X\rightarrow Y})) = \mathbb{1}_X. $$ For any real-valued linear function of $N_{X\rightarrow Y}$, there will always exist a Hermitian operator $H$ for which the value of this linear function is given by $$ \operatorname{Tr}(H J(N_{X\rightarrow Y})). $$ The resulting semidefinite program looks like this: $$ \begin{align} \text{maximize} \quad & \operatorname{Tr}(H P) \\[1mm] \text{subject to} \quad & \operatorname{Tr}_{Y}(P) = \mathbb{1}_X\\[1mm] & P \in \mathrm{Pos}(Y\otimes X), \end{align} $$ where $\mathrm{Pos}(Y\otimes X)$ refers to the set of all positive semidefinite operators acting on $X\otimes Y$.

Like all semidefinite programs, this one has a dual formulation, which is as follows: $$ \begin{align} \text{minimize} \quad & \operatorname{Tr}(Q)\\[1mm] \text{subject to} \quad & \mathbb{1}_Y \otimes Q - H \in \mathrm{Pos}(Y\otimes X)\\[1mm] & Q \in \mathrm{Herm}(X), \end{align} $$ where $\mathrm{Herm}(X)$ is the set of all Hermitian operators acting on $X$.

If you have a specific choice of $H$ in mind, you can solve this optimization problem numerically. I recommend CVX for MATLAB for this purpose.

In general, an analytic solution seems unlikely, but if you correctly guess an optimal solution, you can prove its optimality analytically: using some known facts about semidefinite programming (complelemtary slackness in particular), $N_{X\rightarrow Y}$ can be shown to be optimal if and only if $$ \operatorname{Tr}_{Y}(H J(N_{X\rightarrow Y})) $$ is a Hermitian operator and satisfies $$ \mathbb{1}_Y \otimes \operatorname{Tr}_{Y}(H J(N_{X\rightarrow Y})) - H \in \mathrm{Pos}(Y\otimes X). $$

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    $\begingroup$ Upon further reflection, it seems that your problem may actually be trivial: you can take your channel to be $N_{X\rightarrow Y}(\rho) = \operatorname{Tr}(\rho) |\psi\rangle\langle \psi|$ for $|\psi\rangle$ being an eigenvector of $\sigma_Y$ having the largest possible eigenvalue. I will, however, leave my answer as it is, as it may still be useful in case $\operatorname{Tr}(N_{X\rightarrow Y}(\rho_X)\sigma_Y)$ is replaced by a more interesting linear function of $N_{X\rightarrow Y}$ -- the semidefinite programming approach works for any linear function of $N_{X\rightarrow Y}$. $\endgroup$ – John Watrous Jan 16 at 15:15
  • $\begingroup$ I updated my answer to explain how the semidefinite program looks for a general linear function. $\endgroup$ – John Watrous Jan 16 at 17:02
  • $\begingroup$ Thank you for the very nice answer. The general case is interesting - can I ask how one would find the operator $H$ for a given linear function of $N$? $\endgroup$ – user1936752 Jan 16 at 18:03
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    $\begingroup$ You can think of the existence of a suitable $H$ as being analogous to the fact that linear functions from vectors to scalars always look like an inner product with a fixed vector. If you have a particular linear function in mind, you can compute $H$ by thinking about the standard bases. In the special case that your linear function is the value $\langle c| N_{X\rightarrow Y}(|a\rangle \langle b|) | d\rangle$, for some choice of standard basis states $|a\rangle$ and $|b\rangle$ of $X$ and $|c\rangle$ and $|d\rangle$ of $Y$, then take $H = |d\rangle\langle c| \otimes |b\rangle \langle a|$. $\endgroup$ – John Watrous Jan 16 at 21:10
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    $\begingroup$ Any other linear function can be obtained as a linear combination of these special cases, and so you can take $H$ to be a linear combination of the corresponding operators just described. If your linear function always takes real values, it will be possible to do this in such a way that $H$ is Hermitian. $\endgroup$ – John Watrous Jan 16 at 21:10

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