Leggett-Garg inequalities and the geometry of the cut polytope

TL;DR

This paper reveals a geometric connection between Leggett-Garg inequalities and the cut polytope, introduces new inequalities, and demonstrates novel quantum violations of macrorealism.

Contribution

It establishes a link between Leggett-Garg inequalities and the cut polytope, and presents new inequalities that reveal unexpected quantum violations.

Findings

New family of Leggett-Garg inequalities introduced

Quantum violation of a pentagon inequality shown

Violations occur without violating basic triangle inequalities

Abstract

The Bell and Leggett-Garg tests offer operational ways to demonstrate that non-classical 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 new "pentagon" Leggett-Garg inequality but does not violate any of the basic "triangle" Leggett-Garg inequalities.