Browsing by Author "Kharoof, Aziz"
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Item Open Access Simplicial quantum contextuality(Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2023-05-22) Okay, Cihan; Kharoof, Aziz; İpek, SelmanWe introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine’s theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason’s theorem and Kochen–Specker theorem can be expressed naturally within this new language.Item Open Access Topological methods for studying contextuality: N-Cycle scenarios and beyond(MDPI AG, 2023-07-27) Kharoof, Aziz; İpek, Selman; Okay, CihanSimplicial distributions are combinatorial models describing distributions on spaces of measurements and outcomes that generalize nonsignaling distributions on contextuality scenarios. This paper studies simplicial distributions on two-dimensional measurement spaces by introducing new topological methods. Two key ingredients are a geometric interpretation of Fourier–Motzkin elimination and a technique based on the collapsing of measurement spaces. Using the first one, we provide a new proof of Fine’s theorem characterizing noncontextual distributions in N-cycle scenarios. Our approach goes beyond these scenarios and can describe noncontextual distributions in scenarios obtained by gluing cycle scenarios of various sizes. The second technique is used for detecting contextual vertices and deriving new Bell inequalities. Combined with these methods, we explore a monoid structure on simplicial distributions.