A complete characterization of all-versus-nothing arguments for stabilizer states

Abstract

An important class of contextuality arguments in quantum foundations are the all-versus-nothing (AvN) proofs, generalizing a construction originally due to Mermin. We present a general formulation of AvN arguments and a complete characterization of all such arguments that arise from stabilizer states. We show that every AvN argument for an n-qubit stabilizer state can be reduced to an AvN proof for a three-qubit state that is local Clifford-equivalent to the tripartite Greenberger-Horne-Zeilinger state. This is achieved through a combinatorial characterization of AvN arguments, the AvN triple theorem, whose proof makes use of the theory of graph states. This result enables the development of a computational method to generate all the AvN arguments in Z 2 on n-qubit stabilizer states. We also present new insights into the stabilizer formalism and its connections with logic.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

Towards a complete cohomology invariant for non-locality and contextuality

“…To give an idea of how common this property is, it is sufficient to say that all the models that have appeared in the literature on the sheaf description of contextuality share this property. For instance, it was proven in [ABCP17] that strong contextuality in quantum models obtained by applying Pauli measurements to stabiliser states can be witnessed by only looking at a cycle of size 4.…”

Non-locality, contextuality and valuation algebras: a general theory of disagreement

“…In 2011, Abramsky & Brandenburger introduced an abstract mathematical framework based on sheaf theory to describe these phenomena in a unified treatment, thereby providing a common general theory for the study of non-locality and contextuality, which had been carried out in a rather concrete, example-driven fashion until then [3]. This high-level description shows that contextuality is not a feature specific to quantum theory, but rather a general Notable examples of the sheaf-theoretic notion of contextuality have been found in connection with relational database theory [4], robust constraint satisfaction [5], natural language semantics [6] and logical paradoxes [7][8][9]. This profusion of instances motivates the search for a general theory of contextual semantics, an all-comprehensive approach able to capture the essence and structure of contextual behaviour.…”

“…All rights reserved. Notable examples of the sheaf-theoretic notion of contextuality have been found in connection with relational database theory [4], robust constraint satisfaction [5], natural language semantics [6] and logical paradoxes [7][8][9]. This profusion of instances motivates the search for a general theory of contextual semantics, an all-comprehensive approach able to capture the essence and structure of contextual behaviour.…”

Contextuality degree of quadrics in multi-qubit symplectic polar spaces

“…Other authors have used a correspondence between contextuality and a linear problem initially introduced in [AB11] to provide a global AvN enumeration (see, e.g. [ABCP17]) or links between contextuality and logical paradoxes, as in [Bel64].…”