Measure-independent anomaly of nonlocality

We show that any Bell local state with a hidden nonlocality that can be revealed by local filtering is more, or equally, entangled than nonlocal states. More precisely, it can be deterministically transformed into a nonlocal state by local operations and classical communication. For such a state, there is a clear anomaly of nonlocality for any measures of entanglement and nonlocality. Moreover, we prove that the hidden nonlocality of any bipartite state more, or equally, entangled than nonlocal states can be revealed by local operations and the sending of two one-bit messages, one in each direction. For some particular states, one bit of communication is even enough.