Correlations, Contextuality and Quantum Logic

Davide Fazio, Raffaele Mascella

Patrick A. Heelan

Samson Abramsky, Rui Soares Barbosa

Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

Adel Sohbi, Jaewan Kim

We present a logical type of proof of contextuality for a two-qubit state. We formulate a paradox that cannot be used by a two-qubit system with local measurements to test contextuality while it is possible by using entanglement measurements. With our scheme we achieve ${p}_{\mathrm{Hardy}}\ensuremath{\approx}0.167$. Our approach uses graph theory and the exclusivity principle to give an interpretation of a logical type of proof of quantum correlations. We review the Hardy paradox and find a connection to the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky inequality. We apply the same method to build a paradox based on the Clauser-Horne-Shimony-Holt inequality.

Patrick A. Heelan

Иван Захарович Цехмистро