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 $\rho_B >m$.