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${\cal S}(\sigma)$bethecompactconvexsetofquantumstates$\rho = (\rho_{ij}) \in M_{m\cdot n}$withthefirstpartialtraceequalto$\sigma$, i.e., ${\mathrm{t}\mathrm{r}}_1(\rho )={\rho }_{11}+\cdots +{\rho }_{mm}=\sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \mathcal{S}(\sigma )$ satisfying ${\mathrm{t}\mathrm{r}}_1(\rho )=\sigma$. If $m<n$, there may not be rank one matrix in $\mathcal{S}(\sigma )$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\mathcal{S}$ We also determine ${\rho }^{\ast }\in \mathcal{S}(\sigma )$ with rank bounded by $k$ such that $\mathrm{\parallel }{\mathrm{t}\mathrm{r}}_1{\rho }^{\ast })-\sigma \mathrm{\parallel }$ is minimum for a given unitary similarity invariant norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \mathcal{S}(\sigma )$ is analyzed. Extension of our results and open problems will be mentioned.