Let $X$ and $Y$ be finite dimensional Hilbert spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, $L(X)$ and $L(Y)$ be the sets of linear operators of $X$ and $Y$, $\text{Herm}(X)$ and $\text{Herm}(Y)$ be the set of Hermitian operators of $X$ and $Y$. $Pos(X)$ represents the set of semi-definite elements of $\text{Herm}(X)$. $Φ: L(X)→L(Y)$ is called Hermitian preserving if $Φ(A)$ is always Hermitian for $A\in \text{Herm}(X)$.
We are interested in the following program defined by a tuple $(\Phi,B)$, where:
1.$\Phi: L(X)→L(Y)$ is a Hermiticity-preserving linear map,
2.$B \in \text{Herm}(Y)$.
The pair of optimization problems is as follows:
maximize: rank $Z$
subject to: $\Phi(Z) =B$,
and $Z \in \text{Pos}(X)$.
Is there standard algorithm to solve this problem?
Moreover, we are interested in find some $M$ such that the rank of $M$ equals the value of the above optimization problem.
It is not SDP and it seems much harder than SDP.