Characterising and bounding the set of quantum behaviours in contextuality scenarios

TL;DR

This paper introduces a formal framework and new semidefinite programming techniques to characterize and bound quantum contextual behaviours, revealing insights into quantum advantages in communication and information processing.

Contribution

It develops a novel unitary operator-based relaxation method and applies it to derive bounds and strategies for quantum contextuality scenarios.

Findings

Tight upper bounds on quantum violations of noncontextuality inequalities

Identification of maximally contextual quantum strategies

Demonstration of monogamy of preparation contextuality and secure quantum key distribution

Abstract

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate monogamy of preparation contextuality in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.