Classical vs quantum satisfiability in linear constraint systems modulo an integer

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A system of linear constraints can be unsatisfiable and yet admit a solution in the form of quantum observables whose correlated outcomes satisfy the constraints. Recently, it has been claimed that such a satisfiability gap can be demonstrated using tensor products of generalized Pauli observables in odd dimensions. We provide an explicit proof that no quantum-classical satisfiability gap in any linear constraint system can be achieved using these observables. We prove a few other results for linear constraint systems modulo d > 2. We show that a characterization of the existence of quantum solutions when d is prime, due to Cleve et al, holds with a small modification for arbitrary d. We identify a key property of some linear constraint systems, called phase-commutation, and give a no-go theorem for the existence of quantum solutions to constraint systems for odd d whenever phase-commutation is present. As a consequence, all natural generalizations of the Peres–Mermin magic square and pentagram to odd prime d do not exhibit a satisfiability gap.