Bounds on quantum nonlocality via partial transposition

We explore the link between two concepts: the level of violation of a Bell inequality by a quantum state and discrimination between two states by means of restricted classes of operations, such as local operations and classical communication (LOCC) and separable ones. For any bipartite Bell inequality, we show that its value on a given quantum state cannot exceed the classical bound by more than the maximal quantum violation shrunk by a factor related to distinguishability of this state from the separable set by means of some restricted class of operations. We then consider the general scenarios where the parties are allowed to perform a local pre-processing of many copies of the state before the Bell test (asymptotic and hidden-nonlocality scenarios). We define the asymptotic relative entropy of nonlocality and, for PPT states, we bound this quantity by the relative entropy of entanglement of the partially transposed state. The bounds are strong enough to limit the use of certain states containing private key in the device-independent scenario.