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? 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 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. Control over the state and operators can also be leveraged to create more elaborate protocols for realizing general quantum circuits. In particular, it allows us to establish that a quantum interactive proof system with a classical verifier is as powerful as one with a quantum verifier.