Ranks of quantum states with prescribed reduced states

TL;DR

This paper characterizes the possible ranks of bipartite quantum states with fixed reduced states and applies these results to determine the Choi rank of certain quantum channels.

Contribution

It provides a complete characterization of ranks of bipartite states with prescribed marginals and connects this to the Choi rank of quantum channels with specific properties.

Findings

All possible ranks of bipartite states with given reduced states are determined.

The results are used to find the Choi rank of quantum channels sending the maximally mixed state to a specific state.

The work advances understanding of the structure of quantum states and channels with fixed marginals.

Abstract

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