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Photonics Research

Stronger Hardy-like proof of quantum contextuality

Wen-Rong Qi, Jie Zhou, Ling‐Jun Kong, Zhen‐Peng Xu, Hui-Xian Meng, Rui Liu, Zhou-Xiang Wang, Chenghou Tu, Yongnan Li, Adán Cabello, Jing‐Ling Chen, Hui‐Tian Wang

Published May 9, 2022• 3 citations

Abstract

A Hardy-like proof of quantum contextuality is a compelling way to see the conflict between quantum theory and noncontextual hidden variables (NCHVs), as the latter predict that a particular probability must be zero, while quantum theory predicts a nonzero value. For the existing Hardy-like proofs, the success probability tends to 1/2 when the number of measurement settings n goes to infinity. It means the conflict between the existing Hardy-like proof and NCHV theory is weak, which is not conducive to experimental observation. Here we advance the study of a stronger Hardy-like proof of quantum contextuality, whose success probability is always higher than the previous ones generated from a certain n -cycle graph. Furthermore, the success probability tends to 1 when n goes to infinity. We perform the experimental test of the Hardy-like proof in the simplest case of n = 7 by using a four-dimensional quantum system encoded in the polarization and orbital angular momentum of single photons. The experimental result agrees with the theoretical prediction within experimental errors. In addition, by starting from our Hardy-like proof, one can establish the stronger noncontextuality inequality, for which the quantum-classical ratio is higher with the same n , which provides a new method to construct some optimal noncontextuality inequalities. Our results offer a way for optimizing and enriching exclusivity graphs, helping to explore more abundant quantum properties.

DOI: 10.1364/prj.452704

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A Hardy-like proof of quantum contextuality is a compelling way to see the conflict between quantum theory and noncontextual hidden variables (NCHVs), as the latter predict that a particular probability must be zero, while quantum theory predicts a nonzero value. For the existing Hardy-like proofs, the success probability tends to 1/2 when the number of measurement settings n goes to infinity. It means the conflict between the existing Hardy-like proof and NCHV theory is weak, which is not conducive to experimental observation. Here we advance the study of a stronger Hardy-like proof of quantum contextuality, whose success probability is always higher than the previous ones generated from a certain n -cycle graph. Furthermore, the success probability tends to 1 when n goes to infinity. We perform the experimental test of the Hardy-like proof in the simplest case of n = 7 by using a four-dimensional quantum system encoded in the polarization and orbital angular momentum of single photons. The experimental result agrees with the theoretical prediction within experimental errors. In addition, by starting from our Hardy-like proof, one can establish the stronger noncontextuality inequality, for which the quantum-classical ratio is higher with the same n , which provides a new method to construct some optimal noncontextuality inequalities. Our results offer a way for optimizing and enriching exclusivity graphs, helping to explore more abundant quantum properties.

Contextuality and nonlocality are two fundamental properties of nature. Hardy's proof is considered the simplest proof of nonlocality and can also be seen as a particular violation of the simplest Bell inequality. A fundamental question is: Which is the simplest proof of contextuality? We show that there is a Hardy-like proof of contextuality that can also be seen as a particular violation of the simplest noncontextuality inequality. Interestingly, this new proof connects this inequality with the proof of the Kochen-Specker theorem, providing the missing link between these two fundamental results, and can be extended to an arbitrary odd number n of settings, an extension that can be seen as a particular violation of the n-cycle inequality.

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.

Hardy-type paradoxes provide elegant, inequality-free proofs of quantum contextuality. We introduce a unified logical formalism for these paradoxes, termed logical Hardy-type paradoxes. For any finite quantum scenario of ideal measurements, we prove that the existence of a logical Hardy-type paradox is equivalent to logical contextuality. Specifically, strong contextuality is equivalent to logical Hardy-type paradoxes with success probability SP = 1. These results generalize prior work on (2,k,2), (2,2,d), and n-cycle scenarios. We analyze logical Hardy-type paradoxes in the Mansfield and Klyachko-Can-Binicioglu-Shumovsky (KCBS) scenarios. In the KCBS scenario, we show that there is exactly one type of logical Hardy-type paradox, achieving SP\approx 10.56% for a specific parameter setting.

The violation of contextuality inequalities shows a strong conflict between the noncontextual hidden variable theory and quantum mechanics. A single qutrit is the simplest indivisible quantum system that can allow a state-dependent contextuality test with five observables and a state-independent contextuality test with 13 observables. For any qutrit states, except for the maximally mixed state, one can always find a set of nine observables for which the corresponding inequality is violated [P. Kurzynski and D. Kaszlikowski, Phys. Rev. A 86, 042125 (2012)]. In this paper, we demonstrate the violation of this inequality using single-photon experiments. Our experiments not only enrich the tests of quantum contextuality, but also deepen the understanding of the physical nature of quantum mechanics.

Contextuality is one way of capturing the nonclassicality of quantum theory.The contextual nature of a theory is often witnessed via the violation of noncontextuality inequalities-certain linear inequalities involving probabilities of measurement events.Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [Cabello et al.Phys. Rev. A 88, 032104 (2013); Chudnovsky et al.Ann. Math. 164, 51 (2006)] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either cyclically or anticyclically exclusive.Thus, the noncontextuality inequalities the underlying exclusivity structure of which is as stated, either cyclic or anticyclic, are fundamental to quantum theory.We show that there is a unique noncontextuality inequality for each nontrivial cycle and anticycle.In addition to the foundational interest, we expect this to aid the understanding of contextuality as a resource to quantum computing and its applications to local self-testing.