On the rank of extremal marginal states

TL;DR

This paper investigates the maximum possible rank of extremal states with fixed marginals in quantum systems, confirming the tightness of known bounds in various matrix algebras using advanced mathematical techniques.

Contribution

It extends previous results by providing a positive answer to Rudolph's question on the tightness of the rank bound in multiple matrix algebras.

Findings

Confirmed the tightness of the rank bound in various matrix algebras.

Utilized Choi-Jamiołkowski isomorphism and tensor product methods.

Extended the understanding of extremal marginal states in quantum information theory.

Abstract

Let $\rho_1$ and $\rho_2$ be two states on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}(\rho_1,\rho_2)$, is the set of all states $\rho$ on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ with partial traces $\rho_1, \rho_2$. K. R. Parthasarathy established that if $\rho$ is an extreme point of $\mathcal{C}(\rho_1,\rho_2)$, then the rank of $\rho$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras $\mathbb{M}_3$ and $\mathbb{M}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamio\l kowski isomorphism and tensor product of extreme points.