Device-independent characterizations of a shared quantum state independent of any Bell inequalities

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 {that} a quantum state with minimal size that is able to produce this correlation can always be pure. In this work, we first exhibit two device-independent 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 can be tight, and have various applications in quantum tasks. First, one of the characterizations allows us to bound the entanglement between Alice and Bob using Renyi entropies and also to {bound} the underlying Hilbert space dimension. Second, when the {Hilbert space dimension bound} is tight, the shared pure quantum state has to be maximally entangled. Third, the second characterization gives a sufficient condition that a Bell correlation cannot be generated by particular quantum states. We also show that our results can be generalized to the case of shared mixed states.