Many-body contextuality and self-testing quantum matter via nonlocal games

Abstract: Contextuality is arguably the fundamental property that makes quantum mechanics different from classical physics. It is responsible for quantum computational speedups in both magic-state-injection-based and measurement-based models of computation, and can be directly probed in a many-body setting by multiplayer nonlocal quantum games. Here, we discuss a family of games that can be won with certainty when performing single-site Pauli measurements on a state that is a codeword of a Calderbank-Shor-Steane (CSS) error-correcting quantum code. We show that these games require deterministic computation of a code-dependent Boolean function, and that the classical probability of success is upper bounded by a generalized notion of nonlinearity/nonquadraticity. This success probability quantifies the state's contextuality, and is computed via the function's (generalized) Walsh-Hadamard spectrum. To calculate this, we introduce an efficient, many-body-physics-inspired method that involves identifying the symmetries of an auxiliary hypergraph state. We compute the classical probability of success for several paradigmatic CSS codes and relate it to both classical statistical mechanics models and to strange correlators of symmetry-protected topological states. We also consider CSS submeasurement games, which can only be won with certainty by sharing the appropriate codeword up to local isometries. These games therefore enable self-testing, which we illustrate explicitly for the 2D toric code. We also discuss how submeasurement games enable an extensive notion of contextuality in many-body states.