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Bell-nonlocality quantifiers and their persistent mismatch with the entropy of entanglement

Ari Patrick1, Giulio Camillo1, Fernando Parisio2,*, and Barbara Amaral1,†

  • 1Department of Mathematical Physics, Institute of Physics, University of São Paulo, Rua do Matão 1371, São Paulo 05508-090, São Paulo, Brazil
  • 2Departamento de Física, Universidade Federal de Pernambuco, Recife 50670-901, Pernambuco, Brazil

  • *fernando.parisio@ufpe.br
  • †barbara_amaral@usp.br
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Phys. Rev. A 107, 042410 – Published 7 April, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.042410

Abstract

For some commonly used nonlocality quantifiers, the maximally nonlocal state may not coincide with the maximally entangled state, what became known as the anomaly of nonlocality. More recently, nonanomalous quantifiers have been defined, for which maximal entanglement and maximal nonlocality do coincide. In this work we investigate in detail how these quantifiers behave for nonmaximal-resource states and show that some mismatch remains between the studied figures of merit. Besides the nonlocal volume (a quantifier that counts the volume of the set of behaviors accessible to a specific state that gives rise to nonlocality), we present an alternative quantity referring to states and based on behaviors, the trace-weighted nonlocal volume. The construction is based on the nonlocal volume integral, but weighted by a quantifier of nonlocality for behaviors (the shortest distance between the behavior and the local set). Although in all investigated scenarios the anomaly is not present, the nonlocality quantifiers as compared to the entropy of entanglement are inversely related for some set of states within some Bell scenarios. In the investigated situations this phenomenon occurs when the considered states go through a change of rank. This fact is discussed in general and illustrated through computations for the (2,3,2) scenario.

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Phys. Rev. A 107, 042410– Published 7 April, 2023
  • Received 31 January 2023
  • Accepted 30 March 2023
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