The rank of contextuality

TL;DR

This paper introduces the rank of contextuality (RC), a new measure quantifying the complexity of quantum contextual behaviors, with properties aligning with resource theory and applications in quantum communication.

Contribution

It defines RC as the minimal number of non-contextual behaviors needed to simulate a given contextual behavior, establishing its theoretical properties and connections to hypergraph arboricity.

Findings

RC is a natural measure satisfying faithfulness, monotonicity, and additivity.

Constructed examples demonstrate RC's arbitrary values and its relation to hypergraph arboricity.

The measure applies to various research areas in quantum information.

Abstract

Quantum contextuality is one of the most recognized resources in quantum communication and computing scenarios. We provide a new quantifier of this resource, the rank of contextuality (RC). We define RC as the minimum number of non-contextual behaviors that are needed to simulate a contextual behavior. We show that the logarithm of RC is a natural contextuality measure satisfying several properties considered in the spirit of the resource-theoretic approach. The properties include faithfulness, monotonicity, and additivity under tensor product. We also give examples of how to construct contextual behaviors with an arbitrary value of RC exhibiting a natural connection between this quantifier and the arboricity of an underlying hypergraph. We also discuss exemplary areas of research in which the new measure appears as a natural quantifier.