For a quantum state represented as an $nimes n$ density matrix $sigma in M_n$, let $cS(sigma)$ be the compact convex set of quantum states $ho = (ho_{ij}) in M_{mcdot n}$ with the first partial trace equal to $sigma$, i.e., $r_1(ho) =ho_{11} + cdots + ho_{mm} = sigma$. It is known that if $mge n$ then there is a rank one matrix $ho in cS(sigma)$ satisfying $r_1(ho) = sigma$. 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 $ho^* in cS(sigma)$ with rank bounded by $k$ such that $|r_1(ho^*) - sigma|$ is minimum for a given unitary similarity invariant norm $|cdot|$. Furthermore, the relation between the eigenvalues of $sigma$ and those of $ho in cS(sigma)$ is analyzed. Extension of the results and open problems will be mentioned.
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机译：对于表示为M_n $中$ n n倍密度矩阵$ sigma 的量子态，令$ cS（ sigma）$是量子态的紧凑凸集$ rho =（ rho_ {ij} ） in M_ {m cdot n} $，第一个部分迹线等于$ sigma $，即$ tr_1（ rho）= rho_ {11} + cdots + rho_ {mm} = sigma $。已知如果$ m ge n $则存在一个满足$ tr_1（ rho）= sigma $的秩矩阵$ rho in cS（ sigma）$。如果$ m 展开▼
