  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.
