Restriction on the rank of marginals of bipartite pure states

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 ${\rho }_B>m$.