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

• URL: http://arxiv.org/abs/2605.26920v1
• Date: Tue, 26 May 2026 12:18:51 GMT
• Title: Extremal Marginal States of Maximal Rank in $(d, d+m)$
• Abstract summary: We study the extreme points of the convex set $mathcalC(_2)$ of bipartite quantum states with fixed marginals.<n>We construct extreme points in $(d,,d+m)$ dimension, of rank $d+m$, matching the highest possible value.<n>This proves the existence of extremal states with relatively large rank and also covers all the known examples.
• Score: 0.42970700836450487
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the extreme points of the convex set $\mathcal{C}(ρ_1,ρ_2)$ of bipartite quantum states with fixed marginals $ρ_1$ and $ρ_2$. We construct extreme points in $(d,\,d+m)$ dimension, of rank $d+m$, matching the highest possible value, for all $d\geq 3$, $m > \frac{d^2-2d-2}{2}$ (when $d=2$, $m\geq 1$). 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 $\mathcal{C}(ρ_1,ρ_2)$, 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 $d_1\leq d_2$. Thus, our results show that apart from possibly a few finite cases, for each $d_1$, the maximal rank is achieved almost all times.

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