Maximising linear function over a specific convex set of density matrices

All matrices being discussed in this question are density matrices, so they have the following properties:

• Hermitian
• Positive Semidifinite
• Trace = 1

We are currently in the space of all 4*4 density matrices.

Within this, there is a convex set that is constrained in the following way:

All states that have non-negative quantum conditional entropy. $$Constraint: Tr(Tr_A(\rho) log(Tr_A(\rho))) - Tr(\rho log \rho)) \geq 0$$ Where $$Tr_A$$ refers to the partial trace of $$\rho$$.

The objective function $$Tr(\rho \sigma)$$ where $$\sigma$$ is a state outside this set needs to be maximised over this set. This is a linear function.

$$\sigma$$ is some given state and $$\rho$$ is the only variable.

• I had to look up partial trace en.wikipedia.org/wiki/Partial_trace, and have not digested it yet. Have you shown that $Tr(Tr_A(\rho) log(Tr_A(\rho))) - Tr(\rho log \rho)) \geq 0$ is a convex constraint? If so, can you relate to quantum entropy, quantum relative entropy and.or trace of matrix log, and listed under "Functions and sets" at github.com/hfawzi/cvxquad ? If so, perhaps this could be handled by CVXQUAD under CVX under MATLAB. – Mark L. Stone May 16 at 0:23
• I'm still confused as to your actual optimization problem statement. How does $\sigma$ being outside the set come into play? You need to more clearly state the entirety of the problem: what the decision (optimization) variables are (just $\rho$?), what the objective function is, what all the constraints are, and what the input data consists of. Anyhow, it seems your title is misleading, because you are not talking about an arbitrary convex set, rather, a very specific set of constraints. – Mark L. Stone May 16 at 0:27