On the rank of extremal marginal states

2024/12/13 by Devendra, Repana, Dey, Pankaj, Dey, Santanu
#46L30 #46M05 #46N50 #47L07 #81P47 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2412.10041

Abstract

Let ρ₁ and ρ₂ be two states on ℂ^d₁ and ℂ^d₂ respectively. The marginal state space, denoted by C(ρ₁,ρ₂), is the set of all states ρ on ℂ^d₁⊗ ℂ^d₂ with partial traces ρ₁, ρ₂. K. R. Parthasarathy established that if ρ is an extreme point of C(ρ₁,ρ₂), then the rank of ρ does not exceed √(d₁²+d₂²-1). 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 \mathbbM₃ and \mathbbM₄. 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-Jamiołkowski isomorphism and tensor product of extreme points.

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