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arXiv · 2210.10186

Mermin polytopes in quantum computation and foundations

Cihan Okay·Ho Yiu Chung·Selman Ipek
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Abstract

Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_\beta$ obtained from the Mermin scenario, parametrized by a function $\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $\beta$. 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 $\text{MP}_0$ 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. $\text{MP}_1$ can be seen as a nonlocal toy version of $\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\Lambda$-polytope using $\text{MP}_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.

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Cihan Okay, Ho Yiu Chung, Selman Ipek. 2022-10-18. Mermin polytopes in quantum computation and foundations. https://arxiv.org/abs/2210.10186

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