The nonlocality of certain quantum states can be revealed by using local filters before performing a standard Bell test. This phenomenon, known as hidden nonlocality, has been so far demonstrated only for a restricted class of measurements, namely, projective measurements. Here, we prove the existence of genuine hidden nonlocality. Specifically, we present a class of two-qubit entangled states, for which we construct a local model for the most general local measurements, and show that the states violate a Bell inequality after local filtering. Hence, there exist entangled states, the nonlocality of which can be revealed only by using a sequence of measurements. Finally, we show that genuine hidden nonlocality can be maximal. There exist entangled states for which a sequence of measurements can lead to maximal violation of a Bell inequality, while the statistics of nonsequential measurements is always local.

Quantum steering can be detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Here we discuss the converse problem, namely ensuring that an entangled state is unsteerable, and hence Bell local. We present a simple criterion, applicable to any two-qubit state, which guarantees that the state admits a local hidden state model for arbitrary projective measurements. Specifically, we construct local hidden state models for a large class of entangled states, which can thus not violate any steering or Bell inequality. In turn, this leads to sufficient conditions for a state to be only one-way steerable, and provides the simplest possible example of one-way steering. Finally, by exploiting the connection between steering and measurement incompatibility, we give a sufficient criterion for a continuous set of qubit measurements to be jointly measurable.

Constructing local hidden variable (LHV) models for entangled quantum states is a fundamental problem, with implications for the foundations of quantum theory and for quantum information processing. It is, however, a challenging problem, as the model should reproduce quantum predictions for all possible local measurements. Here we present a simple method for building LHV models, applicable to any entangled state and considering continuous sets of measurements. This leads to a sequence of tests which, in the limit, fully captures the set of quantum states admitting a LHV model. Similar methods are developed for local hidden state models. We illustrate the practical relevance of these methods with several examples.

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the twoqubit Werner state ρ = v |ψ − ψ − | + (1 − v)1/4 via a local hidden variable (LHV) model, where |ψ − denotes the singlet state. We show analytically that these correlations are local for v = 999 × 689 × 10 −6 cos 4 (π/50) 0.6829. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant K G (3) ≤ 1/v 1.4644. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v 0.4553. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.A quantum Bell experiment consists of two (or more) distant observers performing local measurements on a shared entangled quantum state. Remarkably the predictions of quantum theory are here incompatible with a natural definition of locality formulated by Bell [1]. Specifically, the statistics of certain quantum Bell experiments are found to be nonlocal (in the sense of Bell), as witnessed via violation of Bell inequalities. This phenomenon, referred to as quantum nonlocality, represents a fundamental aspect of quantum theory as well as a central resource for quantum information processing [2].Understanding the exact relation between entanglement and quantum nonlocality is a central problem in the foundations of quantum theory, with implications for quantum information processing. While the use of an entangled state is necessary for observing quantum nonlocal correlations, it is interesting to ask if the converse link also holds. That is, can any entangled state lead to a Bell inequality violation, when performing a set of (judiciously chosen, and possibly infinitely many) local measurements? For pure entangled states, the answer turns out to be positive [3]. For mixed entangled states, the situation is more complex, as first discovered by Werner [4], who presented a class of entangled quantum states (now referred to as Werner states) which admit a local hidden variable (LHV) model for any possible local projective measurements. Therefore such states, while being entangled-i.e. inseparable at the level of the Hilbert space-can never lead to nonlocal correlations. Notably, while Werner's original model focused on projective measurements, Barrett [5] presented a LHV model considering the most general non-sequential measurements, i.e. POVMs. These early results triggered much interest, and subsequent works presented various classes of entangled states admitting LHV models [6][7][8][9][10][11][12][13], including results for the multipartite case [14][15][16]; see [17] for a recent review. More sophisticated Bell scenarios have also been ex-1 arXiv:1609.06114v4 [quant-ph]

A sequential steering scenario is investigated, where multiple Bobs aim at demonstrating steering using successively the same half of an entangled quantum state. With isotropic entangled states of local dimension d, the number of Bobs that can steer Alice is found to be N Bob ∼ d/ log d, thus leading to an arbitrary large number of successive instances of steering with independently chosen and unbiased inputs. This scaling is achieved when considering a general class of measurements along orthonormal bases, as well as complete sets of mutually unbiased bases. Moreover, we show that similar results can be obtained in an anonymous sequential scenario, where none of the Bobs know their position in the sequence. Finally, we briefly discuss the implication of our results for sequential tests of Bell nonlocality.