Physicists experimentally verify the multipartite generalized Hardy's paradox

January 15, 2019

In a recent article published in Science Bulletin [1], a joint team led by Professors Jian-Wei Pan, Chao-Yang Lu and Nai-Le Liu at the University of Science and Technology of China, and Jing-Ling Chen at Nankai University, has for the first time verified the multipartite Hardy's paradox in experiment. The researchers have further confirmed Bell nonlocality with the use of Hardy's inequality. Within the experimental errors, the experimental results are in agreement with theoretical predictions given in Ref. [2].

Foundations in quantum mechanics are important issues that researchers have been concentrated on Refs. [3-5]. There exist many paradoxes in quantum mechanics, among which Hardy's paradox [6,7] uses a logical contradiction to prove Bell nonlocality, requiring less times of measurement than those of using Bell's inequality. Bell proposed an idea of a theorem, in the form of a mathematical inequality, that any local hidden-variable theory is incompatible with quantum mechanics. For testing whether a quantum state has Bell nonlocality, there are usually two ways to do so: one with proofs using inequalities, like Bell's inequality, the other with proofs without the use of inequalities, such as the "all-versus-nothing" criteria exemplified by Hardy's theorem [6,7] and the GHZ theorem [8]. The GHZ theorem, which is applicable to more-than-two-party systems, is the first Bell's theorem without inequalities. In its argument, quantum mechanics gives a "-1" result but classical theories give "+1", showing a sharp contradiction between these two types of theories. In 2000, the three-party GHZ theorem was first verified in experiment by Jian-Wei Pan and his colleagues [9].

In 1992 Lucien Hardy, then 19 years old (now holding a position at the Perimeter Institute in Canada), proposed an all-versus-nothing criterion (i.e., Hardy's theorem) that applies to the two-party situation. The two-party Hardy's paradox has been experimentally verified in experiments (see Ref. [10] for an example). In contrast with the GHZ paradox, Hardy's original paradox has a successful probability just around 9% [7]. Generalizing the paradox to multipartite situations can not only increase the successful probability, but also yield Hardy's inequalities that have fairly desired mathematical properties. For instance, Cereceda [11] wrote down in 2004 a first generalized N-party Hardy's paradox, showing that the maximal successful probability can reach 12.5%. He further derived out the corresponding N-party Hardy's inequality, using which Sixia Yu and his colleagues [12] analytically proved the Gisin theorem: Quantum entanglement is equivalent to Bell nonlocality for arbitrary N-party pure states. In 2018, Jing-Ling Chen and his colleagues [2] constructed a general framework for N-party generalized Hardy's paradox. The maximal successful probability can reach 25%. The scheme and experimental setup for generalized Hardy's paradox are shown in Fig. 1. The present experiment has for the first time verified the three- and four-party generalized Hardy's paradoxes in experiment.
[1] Luo YH, Su HY, Huang HL, et al. Experimental Test of Generalized Hardy's Paradox. Sci Bull, 2018,63(24)1611-1615, doi: 10.1016/j.scib.2018.11.025

[2] Jiang SH, Xu ZP, Su HY et al. Generalized Hardy's Paradox. Phys Rev Lett 2018, 120: 0403.

[3] Zhou ZY, Zhu ZH, Liu SL, et al. Quantum twisted double-slits experiments: confirming wavefunctions' physical reality. Sci Bull 2017, 62: 1185-1192.

[4]Long GL. What is the wave function in quantum mechanics? Sci Bull 2017, 62: 1355-1356.

[5]Sanz AS. Bohm's approach to quantum mechanics: alternative theory or practical picture? Front Phys 2019, 14: 11301.

[6] Hardy L. Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories. Phys Rev Lett 1992, 68: 2981.

[7] Hardy L. Nonlocality for two particles without inequalities for almost all entangled states. Phys Rev Lett, 1993, 71: 1665.

[8] Greenberger DM, Horne MA, Shimony A, et al. Bell's theorem without inequalities. Am J Phys, 1990, 12: 1131-1143.

[9] Pan JW, Bouwmeester D, Daniell M. Experimental test of quantum nonlocality in three-photon GHZ entanglement. Nature, 2000, 403: 515-519.

[10] For example, see Torgerson, J. R., et al., "Experimental demonstration of the violation of local realism without Bell inequalities." Physics Letters A 204.5(1995):323-328.

[11] Cereceda JL. Hardy's nonlocality for generalized N-particle GHZ states. Phys Lett A, 2004, 327: 433-437.

[12] Yu SX, Chen Q, Zhang CJ et al., All entangled pure states violate a single Bell's inequality. Phys Rev Lett 2012, 109: 120402.

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