The cohomological and the resource-theoretic perspective on quantum contextuality: common ground through the contextual fraction

TL;DR

This paper unifies cohomological and resource-theoretic approaches to quantum contextuality through the concept of the contextual fraction, providing new invariants and bounds relevant for quantum computation and logical inequalities.

Contribution

It introduces cohomological invariants as witnesses of contextuality and links the contextual fraction to bounds on classical computation costs and logical inequalities.

Findings

Cohomological invariants serve as witnesses of state-dependent contextuality.

Refinements of logical contextuality inequalities are derived.

Upper bounds on classical costs of Boolean functions are established based on the contextual fraction.

Abstract

We unify the resource-theoretic and the cohomological perspective on quantum contextuality. At the center of this unification stands the notion of the contextual fraction. For both symmetry and parity based contextuality proofs, we establish cohomological invariants which are witnesses of state-dependent contextuality. We provide two results invoking the contextual fraction, namely (i) refinements of logical contextuality inequalities, and (ii) upper bounds on the classical cost of Boolean function evaluation, given the contextual fraction of the corresponding measurement-based quantum computation.