Bounding and simulating contextual correlations in quantum theory

TL;DR

This paper develops a hierarchy of semidefinite relaxations to bound and simulate quantum contextual correlations, revealing new maximal violations, resource requirements, and simulation costs in quantum and operational theories.

Contribution

It introduces a versatile hierarchy of relaxations for bounding quantum contextuality and analyzes the resource and simulation costs in quantum and classical models.

Findings

Bound the magnitude of quantum contextuality using the hierarchy.

Determine maximal quantum violations of previously unknown inequalities.

Show mixed states are essential resources for contextuality.

Abstract

We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalised contextuality scenarios. This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality. To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown. We then go further and use it to prove that certain preparation-contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextuality. In the second part of the paper, we turn our attention to the simulation of preparation-contextual correlations in general operational theories. We introduce the information cost of simulating preparation contextuality, which quantifies the additional, otherwise forbidden, information required to simulate contextual correlations in either classical or quantum models. In both cases, we show that the simulation cost can be efficiently bounded using a variant of our hierarchy of semidefinite relaxations, and we calculate it exactly in the simplest contextuality scenario of parity-oblivious multiplexing.