General conditions for maximal violation of non-contextuality in discrete and continuous variables

TL;DR

This paper establishes general spectral conditions for observables that lead to maximal, state-independent violations of non-contextuality inequalities in quantum mechanics, unifying and extending existing frameworks for both discrete and continuous variables.

Contribution

It identifies spectral conditions for observables that demonstrate state-independent contextuality, unifying previous approaches and revealing new scenarios for maximal violation.

Findings

Unifies existing strategies for maximal violation of non-contextual inequalities.

Identifies conditions on spectral decomposition for state-independent contextuality.

Shows impossibility of maximal violation in odd-dimensional Peres-Mermin scenarios.

Abstract

The contextuality of quantum mechanics, i.e. the measurement outcome dependence upon previously made measurements, can be shown by the violation of inequalities based on measurements of well chosen observables. An important property of such observables is that their expectation value can be expressed in terms of probabilities of obtaining two exclusive outcomes. In order to satisfy this, inequalities have been constructed using either observables with a dichotomic spectrum or using periodic functions obtained from displacement operators in phase space. Here we identify the general conditions on the spectral decomposition of observables demonstrating state independent contextuality of quantum mechanics. As a consequence, our results not only unify existing strategies for maximal violation of state independent non-contextual inequalities but also lead to new scenarii enabling such violation. Among the consequences of our results is the impossibility of having a state independent maximal violation of non-contextuality in the Peres-Mermin scenario with discrete observables of odd dimensions.