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arXiv:1411.6882 · Physical Review A

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

Some Sankar Bhattacharya, Arup Roy, Amit Mukherjee, Ramij Rahaman

Published Jul 15, 2015• 1 citation
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Abstract

Hardy's is one of the simplest arguments concerning nonlocality. Recently, Chen et al. [Phys. Rev. A 88, 062116 (2013)] have proposed a more generalized Hardy-like argument and have shown that the probability of success increases with the local system's dimension. Here we study the same in a minimally constrained theory, namely, the generalized no-signaling theory(GNST). We find that not only does the probability of success of this argument increase with the local system dimension in GNST, but it also takes a very simple functional form.

DOI
10.1103/physreva.92.012111

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Hardy's is one of the simplest arguments concerning nonlocality. Recently, Chen et al. [Phys. Rev. A 88, 062116 (2013)] have proposed a more generalized Hardy-like argument and have shown that the probability of success increases with the local system's dimension. Here we study the same in a minimally constrained theory, namely, the generalized no-signaling theory(GNST). We find that not only does the probability of success of this argument increase with the local system dimension in GNST, but it also takes a very simple functional form.
The amount of nonlocality in quantum theory is limited compared to that allowed in generalized no-signaling theory [S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994)]. This feature, for example, gets manifested in the amount of Bell inequality violation as well as in the degree of success probability of Hardy's (Cabello's) nonlocality argument. Physical principles like information causality and macroscopic locality have been proposed for analyzing restricted nonlocality in quantum mechanics, viz. explaining the Cirel'son bound. However, these principles are not very successful in explaining the maximum success probability of Hardy's as well as Cabello's argument in quantum theory. Here we show that a recently proposed physical principle, namely local orthogonality, does better by providing a tighter upper bound on the success probability for Hardy's nonlocality. This bound is relatively closer to the corresponding quantum value compared to the bounds achieved from other principles.
Recently, an alternative form of Hardy's paradox was introduced for two-setting high-dimensional systems [Phys. Rev. A 88, 062116 (2013)], for which the maximum probability of the nonlocal event was shown to increase as the dimension goes larger. Here, we generalize the result to a general scenario with multisetting high-dimensional systems. The general Hardy's paradox (i) reduces to the one by Chen et al. [Phys. Rev. A 88, 062116 (2013)] for two settings, (ii) is equivalent to the ladder proof of nonlocality without inequalities given by Boschi et al. [Phys. Rev. Lett. 79, 2755 (1997)] for two-dimensional systems, and (iii) increases the maximum probability of the nonlocal event for, e.g., three-dimensional systems. In particular, the maximum probability for the five-setting three-dimensional system is increased to 0.401840.401\phantom{\rule{0.16em}{0ex}}840.40184, which is larger than 0.171 originally by Cabello [Phys. Rev. A 58, 1687 (1998)].
Hardy's non-locality theorem for multiple two-level systems is explored in the context of generalized non-local theory. We find non-local but non-signaling probabilities satisfying Hardy's argument for two two-level and three two-level systems. Maximum probability of success of Hardy's argument is obtained for three two-level systems in quantum theory as well as in a more generalized theory. Interestingly, the maximum in the generalized non-local theory for both the two two-level systems and three two-level systems turns out to be the same.
Recently, the principle of nonviolation of information causality [Nature 461, 1101 (2009)] has been proposed as one of the foundational properties of nature. We explore the Hardy's nonlocality theorem for two-qubit systems, in the context of generalized probability theory, restricted by the principle of nonviolation of information causality. Applying a sufficient condition for information causality violation, we derive an upper bound on the maximum success probability of Hardy's nonlocality argument. We find that the bound achieved here is higher than that allowed by quantum mechanics but still much less than what the no-signaling condition permits. We also study the Cabello type nonlocality argument (a generalization of Hardy's argument) in this context.
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. Our proof has all the features of Hardy's and adds the only ingredient of the Einstein-Podolsky-Rosen scenario missing in Hardy's proof: It applies to measurements with an arbitrarily large number of outcomes.
Hardy's nonlocality argument, which establishes incompatibility of quantum theory with local-realism, can also be used to reveal the time-nonlocal feature of quantum states. For spin-1/2 systems, the maximum probability of success of this argument is known to be 25%. We show that this maximum remains 25% for all finite-dimensional quantum systems with suitably chosen observables. This enables a test of the quantum properties of macroscopic systems in analogy to the method of Leggett and Garg.