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arXiv:2211.13642 · Results in Physics

Realigned Hardy’s paradox

Shuai Zhao, Qing Zhou, Si-Ran Zhao, Xinyu Xu, Wen‐Zhao Liu, Li Li, Nai-Le Liu, Qiang Zhang, Jing‐Ling Chen, Kai Chen

Published Nov 23, 2023• 0 citations
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Abstract

Hardy’s paradox provides an “All-versus-Nothing” fashion to directly certify that quantum mechanics cannot be completely described by local realistic theory. However, when considering potential imperfections in experiments, like noisy entanglement source and low detection efficiency, the original Hardy’s paradox can only induce a rather small Hardy violation, and is difficult to reveal. To overcome this challenge, we have proposed a realigned Hardy’s paradox, which can dramatically improve the Hardy violation. Furthermore, we have generalized the realigned Hardy’s paradox to arbitrary even n dichotomic measurements. For n=2, 4 and 6 cases, the realigned Hardy’s paradox can achieve Hardy values approximate 0.4140, 0.7734 and 0.8875 respectively for qubit systems, compared with only 0.09 of the original Hardy’s paradox. Meanwhile, the structure of the realigned Hardy’s paradox is simpler and more robust, in the sense that there is only one Hardy condition. We anticipate that the realigned Hardy’s paradox can tolerate more experimental imperfections and help to stimulate more fascinating quantum information applications.

DOI
10.1016/j.rinp.2023.107210

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Hardy’s paradox provides an “All-versus-Nothing” fashion to directly certify that quantum mechanics cannot be completely described by local realistic theory. However, when considering potential imperfections in experiments, like noisy entanglement source and low detection efficiency, the original Hardy’s paradox can only induce a rather small Hardy violation, and is difficult to reveal. To overcome this challenge, we have proposed a realigned Hardy’s paradox, which can dramatically improve the Hardy violation. Furthermore, we have generalized the realigned Hardy’s paradox to arbitrary even n dichotomic measurements. For n=2, 4 and 6 cases, the realigned Hardy’s paradox can achieve Hardy values approximate 0.4140, 0.7734 and 0.8875 respectively for qubit systems, compared with only 0.09 of the original Hardy’s paradox. Meanwhile, the structure of the realigned Hardy’s paradox is simpler and more robust, in the sense that there is only one Hardy condition. We anticipate that the realigned Hardy’s paradox can tolerate more experimental imperfections and help to stimulate more fascinating quantum information applications.
Bell's theorem states that the quantum mechanical description of physical quantities cannot be fully explained by local realistic theories, laying a solid basis for various quantum information applications. Hardy's paradox is celebrated as the simplest form of Bell's theorem concerning its "All versus Nothing" approach to test local realism. However, due to experimental imperfections, existing tests of Hardy's paradox require additional assumptions of the experimental systems, and these assumptions constitute potential loopholes for faithfully testing local realistic theories. Here, we experimentally demonstrate Hardy's nonlocality through a photonic entanglement source. By achieving a detection efficiency of 82.2%, a quantum state fidelity of 99.10%, and applying high-speed quantum random number generators for the measurement setting switching, the experiment is implemented in a loophole-free manner. During 6 h of running, a strong violation of P_{Hardy}=4.646×10^{-4} up to 5 standard deviations is observed with 4.32×10^{9} trials. A null hypothesis test shows that the results can be explained by local realistic theories with an upper bound probability of 10^{-16348}. These testing results provide affirmative evidence against local realism, and establish an advancing benchmark for quantum information applications based on Hardy's paradox.
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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.
Quantum mechanics allows systems to be entangled with each other, which results in stronger than classical correlations. Many methods of identifying entanglement have been proposed over years, most of which are based on violating some statistical inequalities. In this work we extend the idea formed by Hardy, in which entanglement is not identified with the use of statistical inequalities, but by simultaneous satisfaction of certain conditions. We show that this variant of the Hardy paradox relying on marginal probabilities can be resolved only by true multipartite entangled states. Also, the state resolving this paradox for given local measurements is pure and unique in the case of qubit collections.
Hardy’s paradox (equivalently, Hardy’s non-locality or Hardy’s test) [Phys. Rev. Lett. 68, 2981 (1992)] is used to show non-locality without inequalities, and it has been tested several times using optical circuits. We, for the first time, experimentally test Hardy’s paradox of non-locality in superconducting qubits. For practical verification of Hardy’s paradox, we argue that the error-modeling used in optical circuits is not useful for superconducting qubits. So, we propose a new error-modeling for Hardy’s paradox and a new method to estimate the lower bound on Hardy’s probability (i.e., the probability of a specific event in Hardy’s test) for superconducting qubits. Our results confirmed the theory that any non-maximally entangled state of two qubits violates Hardy’s equations; whereas, any maximally entangled state and product state of two qubits do not exhibit Hardy’s non-locality. Further, we point out the difficulties associated with the practical implementation of quantum protocols based on Hardy’s paradox and propose possible remedies. We also propose two performance measures for any two qubits of any quantum computer based on superconducting qubits.