Mermin polytopes in quantum computation and foundations

TL;DR

This paper classifies the vertices of polytopes derived from the Mermin scenario, revealing their structure and connections to quantum contextuality and nonlocality, with implications for quantum computation foundations.

Contribution

It provides a complete classification of vertices of Mermin polytopes and links them to noncontextuality and quantum computation models.

Findings

Vertices of MP_0 are all deterministic.

MP_1 vertices relate to nonlocal toy models.

Decomposition of the Λ-polytope in 2-qubit case.

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.