Hardy's paradox for high-dimensional systems

Abstract

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.

All-versus-nothing violation of local realism from the Hardy paradox under no-signaling

“…Interestingly, the authors have shown numerically that the maximum probability of success increases with the local system's dimension. This new non-locality argument is equivalent to a violation of a tight Bell Inequality [18]. As one might expect this argument reduces to old Hardy's argument [5] for a special case.…”

“…However, this is not the case with quantum correlations. Both sets of conditions (1) & (6) cannot be satisfied by any local-realistic theory (LRT) [5,18]. One can generalize the above conditions (6) by replacing the last zero condition with a non-zero condition…”

“…In [18] the authors have shown that in quantum theory the success probability of relaxed Hardy test (6) surpasses that of the conventional Hardy's test. It has also been shown that the probability of nonlocal events increases with the local system dimension and for high enough dimensions q RH is almost four times higher than q H .…”

“…This result tells that for showing the contradiction of quantum mechanics with the local realism higher dimensional systems give no advantage in experimental implementation of such a test. Keeping this in mind the authors in reference [18] introduce a Hardy like argument which applies to measurements with an arbitrarily large number of outcomes. They have also showed that the success probability of this modified Hardy's paradox increases with increase in the local system's dimension.…”

“…But recently Chen et.al. [18] have formulated Hardy's non-locality argument for measurements that have more than two outcomes in a stronger way. Interestingly, the authors have shown numerically that the maximum probability of success increases with the local system's dimension.…”