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arXiv:1805.04651 · Physical Review A

Hardy's paradox for multisetting high-dimensional systems

Hui-Xian Meng, Jie Zhou, Zhen‐Peng Xu, Hong-Yi Su, Ting Gao, Fengli Yan, Jing‐Ling Chen

Published Dec 3, 2018• 23 citations
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Abstract

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 $0.401\phantom{\rule{0.16em}{0ex}}84$, which is larger than 0.171 originally by Cabello [Phys. Rev. A 58, 1687 (1998)].

DOI
10.1103/physreva.98.062103

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•Sep 28, 2017•0 citations
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 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 paradox is of fundamental importance in quantum information theory. So far, there have been two types of its extensions into higher dimensions: in the first type the maximum probability of nonlocal events is roughly 9% and remains the same as the dimension changes (dimension-independent), while in the second type the probability becomes larger as the dimension increases, reaching approximately 40% in the infinite limit. Here, we (i) give an alternative proof of the first type, (ii) study the situation in which the maximum probability of nonlocal events can also be dimension-independent in the second type and (iii) conjecture how the situation could be changed in order that (ii) still holds.
Characterizing high-dimensional entangled states is of crucial importance in quantum information science and technology. Recent theoretical progress has been made to extend Hardy's paradox into a general scenario with multisetting and multidimensional systems, which enables the maximum probability of nonlocal events to surpass the bound that was limited in Hardy's original test. Hitherto, no experimental verification has been conducted to verify Hardy's paradox, as most of the previous experimental efforts were restricted to two-dimensional systems. Here, based on two-photon high-dimensional orbital angular momentum (OAM) entanglement, we report an experiment to demonstrate Hardy's paradox for multiple settings and multiple outcomes. We demonstrate the paradox for two-setting higher-dimensional OAM subspaces up to d=7d=7d=7, which reveals that the nonlocal events increase with dimension. Furthermore, we showcase the nonlocality with an experimentally recorded probability of 36.77% for a five-setting three-dimensional OAM subspace via entanglement concentration, thus showing a sharper contradiction between quantum mechanics and classical theory.
Recent advancements have expanded Hardy's nonlocality arguments into multisetting and multidimensional systems to enhance quantum correlations. In comparison with Hardy's nonlocality argument, Cabello's nonlocality argument (CNA) emerges as a superior choice for illustrating nonlocal features. An open question persists regarding the potential extension of CNA to arbitrary (k,dk, dk,d) scenarios. Here, we answer this question both in theory and experiment. Theoretically, by utilizing compatibility graphs, we construct a new logical framework for multisetting and multidimensional CNA, demonstrating an increase in the maximum successful probability with setting kkk and dimension ddd. Experimentally, by employing controllable photonic orbital angular momentum entanglement, we exhibit nonlocality with an experimentally recorded probability of 20.29% in the (2, 4) scenario and 28.72% in the (6, 2) scenario. Our work showcases a sharper contradiction between quantum mechanics and classical theory, surpassing the bound limited by the original version.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
Here we present the most general framework for nnn-particle Hardy's paradoxes, which include Hardy's original one and Cereceda's extension as special cases. Remarkably, for any n≥3n\ge 3n≥3 we demonstrate that there always exist generalized paradoxes (with the success probability as high as 1/2n−11/2^{n-1}1/2n−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.