Extremal Marginal States of Maximal Rank in $(d,d+m)$

Abstract

We study the extreme points of the convex set $\mathcal{C}({\rho }_1,{\rho }_2)$ of bipartite quantum states with fixed marginals ${\rho }_1$ and ${\rho }_2$. We construct extreme points in $(d,d+m)$ dimension, of rank $d+m$, matching the highest possible value, for all $d\ge 3$, $m>\frac{d^2-2d-2}{2}$ (when $d=2$, $m\ge 1$). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of $\mathcal{C}({\rho }_1,{\rho }_2)$, it is sufficient to study the special case $\mathcal{C}({\mathcal{D}}_1,{\mathcal{D}}_2)$, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider $d_1\le d_2$. Thus, our results show that apart from possibly a few finite cases, for each $d_1$, the maximal rank is achieved almost all times.

Preliminary draft; 13 pages (Single column); Comments/Suggestions are welcome