(Non-)Contextuality of Physical Theories as an Axiom

TL;DR

This paper generalizes noncontextual inequalities in quantum mechanics, linking their maximum violations to graph-theoretic parameters like the Lovasz theta-function and fractional packing number, revealing broader implications for quantum and probabilistic theories.

Contribution

It introduces a family of noncontextual inequalities associated with graph compatibility structures and relates their quantum violations to graph invariants, extending the understanding of contextuality.

Findings

Quantum violations are given by the Lovasz theta-function.

Probabilistic theories beyond quantum can violate inequalities more.

Bell inequalities can be reformulated as noncontextual inequalities.

Abstract

We show that the noncontextual inequality proposed by Klyachko et al. [Phys. Rev. Lett. 101, 020403 (2008)] belongs to a broader family of inequalities, one associated to each compatibility structure of a set of events (a graph), and its independence number. These have the surprising property that the maximum quantum violation is given by the Lovasz theta-function of the graph, which was originally proposed as an upper bound on its Shannon capacity. Furthermore, probabilistic theories beyond quantum mechanics may have an even larger violation, which is given by the so-called fractional packing number. We discuss in detail, and compare, the sets of probability distributions attainable by noncontextual, quantum, and generalized models; the latter two are shown to have semidefinite and linear characterizations, respectively. The implications for Bell inequalities, which are examples of noncontextual inequalities, are discussed. In particular, we show that every Bell inequality can be recast as a noncontextual inequality a la Klyachko et al.