Mermin polytopes in quantum computation and foundations

Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes MP_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 MP₀ and MP₁ 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 MP₀ 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. MP₁ can be seen as a nonlocal toy version of Λ-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 2-qubit case, we provide a decomposition of the Λ-polytope using MP₁, whose vertices are classified, and the nonsignaling polytope of the (2,3,2) Bell scenario, whose vertices are well-known.