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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.

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  • $\begingroup$ Finite dimensional Hilbert spaces are $\mathbb{R}^n$ and $\mathbb{C}^n$ only. $\endgroup$
    – user64494
    Feb 12, 2018 at 12:22
  • $\begingroup$ @user64494, let it be $\mathbb{C}^n$. $\endgroup$
    – gondolf
    Feb 12, 2018 at 12:42

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