Ranks of quantum states with prescribed reduced states

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Chi-Kwong Li
Yiu-Tung Poon
Xuefeng Wang


Let $\cM_n$ be the set of $n\times n$ complex matrices. In this note, all the possible ranks of a bipartite state in $\cM_m\otimes \cM_n$ with prescribed reduced states in the two subsystems, are determined. The results are used to determine the Choi rank of quantum channels $\Phi: \cM_m \rightarrow \cM_n$ sending $I/m$ to a specific state $\sigma_2 \in \cM_n$.

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