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(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 MP0 and MP1 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 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 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 MP1, whose vertices are classified, and the nonsignaling polytope of the (2, 3, 2) Bell scenario, whose vertices are well-known.

Automated summary

The paper focuses on the study of polytopes MP(beta) obtained from the Mermin square scenario, which are parametrized by a function beta. The main contribution of the paper is the classification of the vertices of two types of polytopes, MP0 and MP1, based on the parity of beta. The paper also describes the associated graph of the polytopes and provides insights into the noncontextual distributions and simulation of universal quantum computation.