Extremal states of qubit-qutrit system with maximally mixed marginals

Abstract

We study extremal elements of the convex set of qubit-qutrit states whose marginals are maximally mixed. In the two qubit case, it is known that every extreme state of such a convex set is a maximally entangled pure state. In qubit-qutrit case, pure states do not exist in the convex set. We construct mixed extreme states of ranks 2 and 3. Second rank extremal state is entangled whereas third rank extreme element is separable. Parthasarathy obtained an upper bound on the rank of extreme states of such a convex set of a bipartite system of n and m dimensions as n2+m2−1. Thus for a qubit-qutrit system, the rank of an extreme element should be less than 12. Since Parthasarathy's bound for two qubit system is 7 and all extreme elements are of rank one, Rudolph posed a question about its tightness. We establish that Parthasarathy's upper bound is tight for qubit-qutrit system.