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.

Einstein-Podolsky-Rosen steering is a form of inseparability in quantum theory commonly acknowledged to be intermediate between entanglement and Bell nonlocality. However, this statement has so far only been proven for a restricted class of measurements, namely, projective measurements. Here we prove that entanglement, one-way steering, two-way steering, and nonlocality are genuinely different considering general measurements, i.e., single round positive-operator-valued measures. Finally, we show that the use of sequences of measurements is relevant for steering tests, as they can be used to reveal "hidden steering."

The problem of separating classical from quantum correlations is in general intractable and has been solved explicitly only in few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of n ≥ 4 observables X 0 , . . . , X n−1 , where each consecutive pair {X i , X i+1 }, sum modulo n, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicioglu-Shumovsky scenarios, which are the simplest ones for, respectively, locality and noncontextuality. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension.