Ranks and eigenvalues of states with prescribed reduced states

For a quantum state represented as an $n\times n$ density matrix $σ\in M_n$, let ${\cal S}(σ)$ be the compact convex set of quantum states $ρ= (ρ_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $σ$, i.e., ${\rm tr}_1(ρ) = ρ_{11} + \cdots + ρ_{mm} = σ$. It is known that if $m \ge n$ then there is a rank one matrix $ρ\in {\cal S}(σ)$ satisfying ${\rm tr}_1(ρ) = σ$. If $m < n$, there may not be rank one matrix in ${\cal S}(σ)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set ${\cal S}$ We also determine $ρ^* \in {\cal S}(σ)$ with rank bounded by $k$ such that $\|{\rm tr}_1ρ^*) - σ\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $σ$ and those of $ρ\in {\cal S}(σ)$ is analyzed. Extension of our results and open problems will be mentioned.

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