Ranks and eigenvalues of states with prescribed reduced states

For a quantum state represented as an $n\times n$ density matrix $\sigma \in M_n$, let $\cS(\sigma)$ be the compact convex set of quantum states $\rho = (\rho_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $\sigma$, i.e., $\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \cS(\sigma)$ satisfying $\tr_1(\rho) = \sigma$. If $m < n$, there may not be any rank one matrix in $\cS(\sigma)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\cS$. We also determine $\rho^* \in \cS(\sigma)$ with rank bounded by $k$ such that $\|\tr_1(\rho^*) - \sigma\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \cS(\sigma)$ is analyzed. Extension of the results and open problems will be mentioned.