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

Cohomological framework for contextual quantum computations

“…This is the known Mermin inequality. Here we demonstrate it invoking the phase function; also see [29].…”

“…Refs. [28], [29] provide symmetry-based contextuality proofs invoking the cohomology of the symmetry group, and represent the left link of Diagram 1. The present work addresses the right link of the diagram (shown as a dashed line) between cohomology and computation, and it uses the same algebraic machinery as [28], [29].…”

“…Refs. [28] and [29] introduced a cohomological framework to study contextuality proofs based on parity and symmetry. Here, we briefly recollect the cohomological structures involved.…”

On Measures and Measurements: a Fibre Bundle approach to Contextuality

“…In refs. [13][14][15][16], the authors point out the influence of cohomology in contextuality and nonlocality. We hope this bundle approach will reforce this relation.…”

“…: a nontrivial homology class implies the nontriviality of some bundle and forbids such extension. In this sense, a nontrivial first homology group says that a scenario allows for contextuality, while a nontrivial cohomology class is a way of witnessing contextuality [13][14][15][16].…”