Quantum states with the same reduced state

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