Self-testing in the compiled setting via tilted-CHSH inequalities

Abstract: In a Bell scenario, a classical verifier interacts with two non-communicating (quantum) provers. To an observer, the behaviour of the provers in this interaction is modeled by correlations. Certain correlations allow the verifier to certify, or self-test, the underlying quantum state and measurements. Self-testing underpins numerous device-independent quantum protocols with a classical verifier, yet, a drawback of using self-tests in applications is the required no-communicating assumption between the provers. To address this issue, Kalai et al.~(STOC '23) introduce a cryptographic procedure which "compiles" these scenarios into a multi-round interaction between a verifier and a single computationally bounded prover. In this work, we formalize a notion of self-testing for compiled two-prover Bell scenarios. In addition, we prove that the quantum value is preserved under compilation for the family of tilted-CHSH inequalities (up to negligible factors). We also show that any maximal violation in the compiled setting of inequalities from this family satisfies a notion of self-testing in the compiled setting. More specifically, we show that maximal violations of these inequalities imply the existence of an efficient isometry that recovers the measurement action on the state after the first round.