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?


1 Answer 1


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 $$ \mathbb{1}_Y \otimes \operatorname{Tr}_{Y}(H J(N_{X\rightarrow Y})) - H \in \mathrm{Pos}(Y\otimes X). $$

  • 2
    $\begingroup$ 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 $N_{X\rightarrow Y}$. $\endgroup$ Commented Jan 16, 2020 at 15:15
  • 1
    $\begingroup$ I updated my answer to explain how the semidefinite program looks for a general linear function. $\endgroup$ Commented Jan 16, 2020 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 $N$? $\endgroup$ Commented Jan 16, 2020 at 18:03
  • 2
    $\begingroup$ 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 $H = |d\rangle\langle c| \otimes |b\rangle \langle a|$. $\endgroup$ Commented Jan 16, 2020 at 21:10
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Jan 16, 2020 at 21:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.