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.
What is the best way to go about this?