In a recent article published in Science Bulletin , 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. .
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 . 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 .
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.  for an example). In contrast with the GHZ paradox, Hardy's original paradox has a successful probability just around 9% . 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  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  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  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.
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