Necessary and sufficient conditions for state-independent measurement contextual scenarios

TL;DR

This paper establishes a complete criterion based on graph theory for identifying measurement scenarios that demonstrate state-independent contextuality in quantum systems, extending beyond traditional paradigms.

Contribution

It provides a necessary and sufficient condition using fractional chromatic number to identify all scenarios exhibiting state-independent contextuality, including new cases beyond Kochen-Specker.

Findings

Contextuality measure depends only on quantum state spectrum.

State-independent contextuality is linked to the maximally mixed state's properties.

Graph-theoretic condition using fractional chromatic number characterizes all such scenarios.

Abstract

The problem of identifying measurement scenarios capable of revealing state-independent contextuality in a given Hilbert space dimension is considered. We begin by showing that for any given dimension $d$ and any measurement scenario consisting of projective measurements, (i) the measure of contextuality of a quantum state is entirely determined by its spectrum, so that pure and maximally mixed states represent the two extremes of contextual behavior, and that (ii) state-independent contextuality is equivalent to the contextuality of the maximally mixed state up to a global unitary transformation. We then derive a necessary and sufficient condition for a measurement scenario represented by an orthogonality graph to reveal state-independent contextuality. This condition is given in terms of the fractional chromatic number of the graph $\chi_f(G)$ and is shown to identify all state-independent contextual measurement scenarios including those that go beyond the original Kochen-Specker paradigm \cite{Yu-Oh}.