Duality of Entanglement Norms

TL;DR

This paper explores duality relationships between four quantum information norms on tensor spaces, revealing new insights into entanglement measures and criteria for separability and Schmidt number.

Contribution

It establishes duality relationships among key norms, providing elementary proofs and generalizations of entanglement criteria in quantum information theory.

Findings

Product numerical radius is dual to robustness of entanglement

S(k)-norm is dual to the projective tensor norm

Generalization of the cross norm criterion to arbitrary Schmidt number

Abstract

We consider four norms on tensor product spaces that have appeared in quantum information theory and demonstrate duality relationships between them. We show that the product numerical radius is dual to the robustness of entanglement, and we similarly show that the S(k)-norm is dual to the projective tensor norm. We show that, analogous to how the product numerical radius and the S(k)-norm characterize k-block positivity of operators, there is a natural version of the projective tensor norm that characterizes Schmidt number. In this way we obtain an elementary new proof of the cross norm criterion for separability, and we also generalize both the cross norm and realignment criteria to the case of arbitrary Schmidt number.