Information-Theoretic and Operational Measures of Quantum Contextuality

TL;DR

This paper develops an information-theoretic framework to quantify quantum contextuality, introducing measures based on mutual information energy and commutator expectations, with applications to spin-1 systems and geometric visualization.

Contribution

It introduces novel, complementary measures of quantum contextuality and connects them to uncertainty relations, providing analytical expressions and geometric insights for spin-1 systems.

Findings

States with optimal uncertainty sum show no operational contextuality.

States with high operational contextuality satisfy a nontrivial Robertson bound.

Closed-form expressions for measures are derived in the KCBS scenario.

Abstract

We propose an information -- theoretic framework for quantifying Kochen-Specker contextuality. Two complementary measures are introduced: the mutual information energy, a state-independent quantity inspired by Onicescu's information energy that captures the geometric overlap between joint eigenspaces within a context; and an operational measure based on commutator expectation values that reflects contextual behavior at the level of measurement outcomes. We establish a hierarchy of bounds connecting these measures to the Robertson uncertainty relation, including spectral, purity-corrected, and operator norm estimates. The framework is applied to the Klyachko-Can-Binicio\u{g}lu-Shumovsky (KCBS) scenario for spin-1 systems, where all quantities admit closed-form expressions. The Majorana-stellar representation furnishes a common geometric platform on which both the operational measure and the uncertainty products can be analyzed. For spin-1, this representation yields a three-dimensional Euclidean-like visualization of the Hilbert space in which, states lying on a plane exhibit maximum uncertainty for the observable along the perpendicular direction; simultaneous optimization across all KCBS contexts singles out a unique state on the symmetry axis. Notably, states achieving the optimal sum of uncertainty products exhibit vanishing operational contextuality, while states with substantial operational contextuality satisfy a nontrivial Robertson bound -- the two extremes are achieved by distinct quantum states.