Restriction on the rank of marginals of bipartite pure states

TL;DR

This paper proves that in bipartite quantum systems, pure states cannot have marginals with certain rank restrictions, specifically when the smaller subsystem's dimension is less than the larger's, extending the result to general $m imes n$ systems.

Contribution

It establishes a fundamental restriction on the existence of pure states with given marginals in bipartite quantum systems, generalizing previous results to arbitrary dimensions.

Findings

No pure state exists in the convex set with fixed marginals when the smaller subsystem's dimension is less than the larger's.

The result applies to arbitrary $ ho_A$ and $ ho_B$ with rank of $ ho_B$ exceeding the smaller subsystem's dimension.

Generalization to all $m imes n$ bipartite systems where $m < n$.

Abstract

Consider a qubit-qutrit ($2 \times 3$) composite state space. Let $C(\{1}{2}I_2, \{1}{3}I_3)$ be a convex set of all possible states of composite system whose marginals are given by $\{1}{2}I_2$ and $\{1}{3}I_3$ in two and three dimensional spaces respectively. We prove that there exists no pure state in $C(\{1}{2}I_2, \{1}{3}I_3)$. Further we generalize this result to an arbitrary $m \times n$ bipartite systems. We prove that for $m < n$, no pure state exists in the convex set $C(\rho_A,\rho_B)$, for an arbitrary $\rho_A$ and rank of $\rho_B >m$.