Browsing by Subject "Algebraic topology"

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    Mermin polytopes in quantum computation and foundations
    (Rinton Press Inc., 2023-06-27) Okay, Cihan; Chung, Ho Yiu; İpek, Selman
    Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes β obtained from the Mermin scenario, parametrized by a function β on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes 0 and 1 depending on the parity of β. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of 0 turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. 1 can be seen as a nonlocal toy version of A-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 2-qubit case, we provide a decomposition of the A-polytope using 1, whose vertices are classified, and the nonsignaling polytope of the (2, 3, 2) Bell scenario, whose vertices are well-known.
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    Topological methods for studying contextuality: N-Cycle scenarios and beyond
    (MDPI AG, 2023-07-27) Kharoof, Aziz; İpek, Selman; Okay, Cihan
    Simplicial 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.