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Physical Review A

Testing Hardy’s nonlocality with n spin-1/2 particles

Guruprasad Kar

Published Jul 1, 1997• 21 citations
Publisher

Abstract

For a system of $n$ spin-$\frac{1}{2}$ particles and for two measurement possibilities for each particle there is always a state showing Hardy's nonlocality. Mixed states of an $n$- $(n>~3)$ particle system can only show Hardy's nonlocality. Product states for any $n$ never show nonlocality.

DOI
10.1103/physreva.56.1023

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For a system of nnn spin-12\frac{1}{2}21​ particles and for two measurement possibilities for each particle there is always a state showing Hardy's nonlocality. Mixed states of an nnn- (n> 3)(n>~3)(n> 3) particle system can only show Hardy's nonlocality. Product states for any nnn never show nonlocality.
In this paper we provide a simple proof of the fact that for a system of two spin-1/2 particles, and for a choice of observables, there is a unique state which shows Hardy-type nonlocality. Moreover, an explicit expression for the probability that an ensemble of particle pairs prepared in such a state exhibits a Hardy-type nonlocality contradiction is given in terms of two independent parameters related to the observables involved. Incidentally, a wrong statement expressed in Mermin's proof of the converse [N.D. Mermin, Am. J. Phys. 62, 880 (1994)] is pointed out.
We critically review earlier works on Hardy's nonlocality argument for two spin-sss systems and show that solutions previously found in this regard were restricted due to imposition of some conditions which have no role in the argument of nonlocality. We provide a compact form of the nonlocality condition for two spin-sss particles, and we also extend it to nnn number of spin-sss particles. Finally we apply a more general kind of nonlocality argument, still without an inequality, to higher-spin systems.
Hardy’s theorem on nonlocality has been verified by a series of experiments with two-qubit entangled pure states. However, in this paper we demonstrate the experimental test of the theorem by using the two-photon entangled mixed states. We first investigate the generic logic in Hardy’s proof of nonlocality, which can be applied for arbitrary two-qubit mixed polarization entangled states and can be reduced naturally to the well-known logic tested successfully by the previous pure state experiments. Then, the optimized violations of locality for various experimental parameters are delivered by the numerical method. Finally, the logic argued above for testing Hardy’s theorem on nonlocality is demonstrated experimentally by using the mixed entangled-photon pairs generated via pumping two type-I BBO crystals. Our experimental results shows that Hardy’s proof of nonlocality can also be verified with two-qubit polarization entangled mixed states, with a violation of about 3.4 standard deviations.