  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.
