We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e. incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggest the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.The correlations resulting from local measurements on an entangled quantum state cannot be explained by a local theory. This aspect of entanglement, termed quantum nonlocality, is captured by two inequivalent notions, namely Bell nonlocality [1, 2] and EPR steering [3][4][5]. The strongest form of this phenomenon is Bell nonlocality, witnessed via the violation of Bell inequalities. Steering represents a strictly weaker form of quantum nonlocality [4], witnessed via violation of steering inequalities [6]. Both aspects have been extensively investigated in recent years, as they play a central role in the foundations of quantum theory and in quantum information processing.Interestingly quantum nonlocality is based on two central features of quantum theory, namely entanglement and incompatible measurements. Specifically, performing (i) arbitrary local measurements on a separable state, or (ii) compatible measurements on an (arbitrary) quantum state can never lead to any form of quantum nonlocality. Hence the observation of quantum nonlocality implies the presence of both entanglement and incompatible measurements. It is interesting to explore the converse problem. Two types of questions can be asked here (see Fig.

Einstein-Podolsky-Rosen steering is a form of quantum nonlocality exhibiting an inherent asymmetry between the observers, Alice and Bob. A natural question is then whether there exist entangled states which are one-way steerable, that is, Alice can steer Bob's state, but it is impossible for Bob to steer the state of Alice. So far, such a phenomenon has been demonstrated for continuous variable systems, but with a strong restriction on allowed measurements, namely, considering only Gaussian measurements. Here we present a simple class of entangled two-qubit states which are one-way steerable, considering arbitrary projective measurements. This shows that the nonlocal properties of entangled states can be fundamentally asymmetrical.

Intensive studies of entanglement properties have proven essential for our understanding of quantum many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems because the available multipartite Bell inequalities involve correlations among many particles, which are difficult to access experimentally. We constructed multipartite Bell inequalities that involve only two-body correlations and show how they reveal the nonlocality in many-body systems relevant for nuclear and atomic physics. Our inequalities are violated by any number of parties and can be tested by measuring total spin components, opening the way to the experimental detection of many-body nonlocality, for instance with atomic ensembles.