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

Abstract

We establish a strong link between two apparently unrelated topics: the study of conflicting information in the formal framework of valuation algebras, and the phenomena of non-locality and contextuality. In particular, we show that these peculiar features of quantum theory are mathematically equivalent to a general notion of disagreement between information sources. This result vastly generalizes previously observed connections between contextuality, relat- ional databases, constraint satisfaction problems and logical paradoxes, and gives further proof that contextual behaviour is not a phenomenon limited to quantum physics, but pervades various domains of mathematics and computer science. The connection allows to translate theorems, methods and algorithms from one field to the other, and paves the way for the application of generic inference algorithms to study contextuality. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

Observers of quantum systems cannot agree to disagree

“…Our setting is appealing in quantum information for its applications to communication complexity, cryptography, teleportation, and many other scenarios. Reference 31 introduces a different notion of disagreement in a no-signaling context. Here, disagreement concerns pieces of information about some variables, and agreement refers to consistency in the information provided about the variables.…”

From the problem of Future Contingents to Peres-Mermin square experiments: An introductory review to Contextuality

“…In fact, as we saw with the Spekkens approach and the CbD one, contextuality has been shown to go beyond the quantum theory. Several publications have shown the relations between the sheaf-theoretic approach of contextuality and phenomena such as relational database theory, robust constraint satisfaction, natural language semantics and logical paradoxes, a general theory of which, called contextual semantics, is given in [205]. Finally, the deeper and more general work in that direction is the formalisation of contextuality as a ressource by Duarte and Amaral [173,206].…”

Classical Logic, Classical Probability, and Quantum Mechanics

“…Thus the complexity of detecting possibilistic complexity runs completely in parallel with the probabilistic case. 6 The "edge of logical contradiction" vs. the "boundary of paradox"…”

“…• The logical perspective on contextuality leads to the recognition that the same structures arise in many non-quantum areas, including databases [1], constraint satisfaction [7], and generic inference [6].…”

“…A Bell inequality establishes a bound for the value of a • v e amongst non-contextual models e. For more general models, this quantity is limited only by the algebraic bound6 a := C∈M max a[ C, s ] | s ∈ O C .The violation of a Bell inequality a, R by a model e is max{0, a • v e − R}. However, it is useful to normalise this value by the maximum possible violation in order to give a better idea of the extent to which the model violates the inequality.…”