A classical leash for a quantum system

Can a classical experimentalist command an untrusted quantum system to realize arbitrary quantum dynamics, aborting if it misbehaves? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. We prove a rigidity theorem for the famous Clauser-Horne-Shimony-Holt (CHSH) game, first formulated to provide a means of experimentally testing the violation of the Bell inequalities. The theorem shows that the only way for the two non-communicating quantum players to win many games played in sequence is if their shared quantum state is close to the tensor product of EPR states (Bell states) and their measurements are the optimal CHSH measurements on successive qubits. This theorem may be viewed as analogous to classical multi-linearity testing, in the sense that the outcome of local checks gives a characterization of a global object.