Quantum state, reduced state, partial trace, quantum channel, rank, eigenvalue.
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 $\sigma_2 \in \cM_n$.
Li, Chi-Kwong; Poon, Yiu-Tung; and Wang, Xuefeng.
"Ranks of quantum states with prescribed reduced states",
Electronic Journal of Linear Algebra,
Volume 34, pp. 331-342.