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Leggett-Garg inequalities and the geometry of the cut polytope

David Avis, Patrick Hayden, and Mark M. Wilde
Phys. Rev. A 82, 030102(R) – Published 23 September 2010

Abstract

The Bell and Leggett-Garg tests offer operational ways to demonstrate that nonclassical behavior manifests itself in quantum systems, and experimentalists have implemented these protocols to show that classical worldviews such as local realism and macrorealism are false, respectively. Previous theoretical research has exposed important connections between more general Bell inequalities and polyhedral combinatorics. We show here that general Leggett-Garg inequalities are closely related to the cut polytope of the complete graph, a geometric object well-studied in combinatorics. Building on that connection, we offer a family of Leggett-Garg inequalities that are not trivial combinations of the most basic Leggett-Garg inequalities. We then show that violations of macrorealism can occur in surprising ways, by giving an example of a quantum system that violates the “pentagon” Leggett-Garg inequality but does not violate any of the basic “triangle” Leggett-Garg inequalities.

  • Figure
  • Received 27 April 2010

DOI:https://doi.org/10.1103/PhysRevA.82.030102

©2010 American Physical Society

Authors & Affiliations

David Avis1,2, Patrick Hayden1, and Mark M. Wilde1

  • 1School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2A7
  • 2Graduate School of Informatics, Kyoto University, 36-1 Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

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Issue

Vol. 82, Iss. 3 — September 2010

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