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.
We propose a genuine multiparty correlation measure for a multiparty quantum system as the trace norm of the cumulant of the state. The legitimacy of our multiparty correlation measure is explicitly demonstrated by proving it satisfies the five basic conditions required for a correlation measure. As an application we construct an efficient algorithm for the calculation of our measures for all stabilizer states.
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.
Our results demonstrated that MSC transplantation could inhibit LV remodeling and improve heart function.
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