Hardy's proof is considered the simplest proof of nonlocality. Here we introduce an equally simple proof that (i) has Hardy's as a particular case, (ii) shows that the probability of nonlocal events grows with the dimension of the local systems, and (iii) is always equivalent to the violation of a tight Bell inequality.PACS numbers: 03.65. Ud, 03.67.Mn, 42.50.Xa Introduction.-Nonlocality, namely, the impossibility of describing correlations in terms of local hidden variables [1], is a fundamental property of nature. Hardy's proof [2,3], in any of its forms [4][5][6][7], provides a simple way to show that quantum correlations cannot be explained with local theories. Hardy's proof is usually considered "the simplest form of Bell's theorem" [8].On the other hand, if one wants to study nonlocality in a systematic way, one must define the local polytope [9] corresponding to any possible scenario (i.e., for any given number of parties, settings, and outcomes) and check whether quantum correlations violate the inequalities defining the facets of the corresponding local polytope. These inequalities are the so-called tight Bell inequalities. In this sense, Hardy's proof has another remarkable property: It is equivalent to a violation of a tight Bell inequality, the Clauser-Horne-ShimonyHolt (CHSH) inequality [10]. This was observed in [5].Hardy's proof requires two observers, each with two measurements, each with two possible outcomes. The proof has been extended to the case of more than two measurements [11,12], and more than two outcomes [13][14][15]. However, none of these extensions is equivalent to the violation of a tight Bell inequality.The aim of this Letter is to show that, if we remove the requirement that the measurements have two outcomes, then Hardy's proof can be formulated in a much powerful way. The new formulation shows that the maximum probability of nonlocal events, which has a limit of 0.09 in Hardy's formulation and previously proposed extensions, actually grows with the number of possible outcomes, tending asymptotically to a limit that is more than four times higher than the original one. Moreover, for any given number of outcomes, the new formulation turns out to be equivalent to a violation of a tight Bell inequality, a feature that suggest that this formulation is more fundamental than any other one proposed previously. All this while preserving the simplicity of Hardy's original proof.A new formulation of Hardy's paradox.-Let us consider two observers, Alice, who can measure either A 1 or A 2 on her subsystem, and Bob, who can measure B 1 or B 2 on his.

Einstein-Podolsky-Rosen steering is a form of quantum nonlocality intermediate between entanglement and Bell nonlocality. Although Schrödinger already mooted the idea in 1935, steering still defies a complete understanding. In analogy to “all-versus-nothing” proofs of Bell nonlocality, here we present a proof of steering without inequalities rendering the detection of correlations leading to a violation of steering inequalities unnecessary. We show that, given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob's normalized conditional states can be regarded as two different pure states provides a criterion for Alice-to-Bob steerability. A steering inequality equivalent to the all-versus-nothing proof is also obtained. Our result clearly demonstrates that there exist many quantum states which do not violate any previously known steering inequality but are indeed steerable. Our method offers advantages over the existing methods for experimentally testing steerability, and sheds new light on the asymmetric steering problem.

We present a study of quantum contextuality of three-dimensional mixed states for the Klyachko-Can-Binicioglu-Shumovsky (KCBS) and the Kurzyński-Kaszlikowski (KK) noncontextuality inequalities. For any class of states whose eigenvalues are arranged in decreasing order, a universal set of measurements always exists for the KK inequality, whereas none does for the KCBS inequality. This difference can be reflected from the spectral distribution of the overall measurement matrix. Our results would facilitate the error analysis for experimental setups, and our spectral method in the paper combined with graph theory could be useful in future studies on quantum contextuality.

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Here we present the most general framework for n-particle Hardy's paradoxes, which include Hardy's original one and Cereceda's extension as special cases. Remarkably, for any n ≥ 3 we demonstrate that there always exist generalized paradoxes (with the success probability as high as 1/2 n−1 ) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardy's inequalities, which enable us to detect Bell's nonlocality for more quantum states.

We propose an improved bound for the difference between phase and bit error rate in measurementdevice-independent quantum key distribution with uncharacterized qubits. We show by simulations that the bound considerably increases the final key rates.

A new theory-independent noncontextuality inequality is presented [Phys. Rev. Lett. 115, 110403 (2015)] based on Kochen-Specker (KS) set without imposing the assumption of determinism. By proposing novel noncontextuality inequalities, we show that such result can be generalized from KS set to the noncontextuality inequalities not only for state-independent but also for state-dependent scenario. The YO-13 ray and n cycle ray are considered as examples.

We derive a set of Bell-type inequalities for arbitrarily high-dimensional systems, based on the assumption of partial separability in the hybrid local-nonlocal hidden variable model. Partially entangled states would not violate the inequalities, and thus upon violation, these Bell-type inequalities are sufficient conditions to detect the full N -particle entanglement and invalidity of the hybrid local-nonlocal hidden variable description.