Device-independent characterizations of the quantum state in a Bell experiment.

In a Bell experiment two parties share a quantum state and perform local measurements on their subsystems separately, and the statistics of the measurement outcomes are recorded as a Bell correlation. For any Bell correlation, it turns out a quantum state with minimal size that is able to produce this correlation can always be pure. In this work, we first exhibit two characterizations for the pure state that Alice and Bob share using only the correlation data. Specifically, we give two conditions that the Schmidt coefficients must satisfy, which have very broad applications in many quantum tasks. First, one of the characterizations allows us to bound the entanglement between Alice and Bob using Renyi entropies and also to estimate the underlying Hilbert space dimension. Second, we provide examples to show that our results can be tight, where the shared quantum state is characterized completely. In this situation, our result serves as a general theoretical tool to self-test bipartite pure quantum states in a device-independent manner. Third, the second characterization gives a sufficient condition that a Bell correlation cannot be generated by maximally entangled states of certain dimension. We also show that our results can be generalized to the case of shared mixed states.