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Journal of Anhui University

A proof of Hardy theorem

XU Fu-xin

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

It is proofed , using two spin 1/2 entangled particles , that quantum mechanics has the nonlocal property which cannot be accounted for by Einstein, Podolsky and Rosen local realism theory, and this proof need not inequalities. The non-classical conclusion of quantum mechanics cannot be recovered by EPR local realism theory.

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Local realism has been knocked down by the experiments with entangled pairs of particles based on Bell's theorem(J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964)). However, there has been continuing debate on whether locality or realism is the problem. In this work, we analyzed the Einstein-Podolsky-Rosen thought experiment of Bohm's version using information theory and thermodynamics. The inference of non-locality from EPR experiments will be against the principle of non-realism of quantum mechanics. Therefore, the experiments about quantum entanglement cannot provide any proof to accuse locality.
The singlet state of two spin-32\frac{3}{2}23​ particles allows a proof of Bell's theorem without inequalities with two distinguishing features: any local observable can be regarded as an Einstein-Podolsky-Rosen element of reality, and the contradiction with local realism occurs not only for some specific local observables but for any rotation whereof.
We use the Greenberger-Horne-Zeilinger [in Bell's Theorem, Quantum Theory,and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989)] approach to present three demonstrations of the failure of Einstein-Podolsky-Rosen (EPR) [Phys. Rev. 47, 777 (1935)] local realism for the case of two spinless particles in a two-particle interferometer. The original EPR assumptions of locality and reality do not suffice for this. First, we use the EPR assumptions of locality and reality to establish that in a two-particle interferometer, the path taken by each particle is an element of reality. Second, we supplement the EPR premises by the postulate that when the path taken by a particle is an element of reality, all paths not taken are empty. We emphasize that our approach is not applicable to a single-particle interferometer because there the path taken by the particle cannot be established as an element of reality. We point out that there are real conceptual differences between single-particle, two-particle, and multiparticle interferometry.
A recent discovery by Hardy [Phys. Rev. Lett. 71, 1665 (1993)] is developed and extended. It shows that the inconsistency between quantum mechanics and Einstein-Podolsky-Rosen assumptions about locality and reality can be demonstrated for two photons or particles with spin 1/2, as in experiments that test Bell inequalities, but without inequalities, as in what Greenberger, Horne, and Zeilinger discovered for three or four particles [in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 74]. Here all the possibilities are found. The measurement choices can be different for the two particles. They are described directly in terms of angles for spin measurements. Hardy's result and its converse are both proved: almost any state will do (with measurement choices made to accommodate) and any choice of two different measurement possibilities for each particle will do (with a state made to accommodate). A simple example is given; it allows the calculations to be done quickly with Pauli matrices.
There is an opinion that the Bohm reformulation of the EPR paradox in terms of spin variables is equivalent to the original one. In this note we show that such an opinion is not justified. We apply to the original EPR problem the method which was used by Bell for the Bohm reformulation. He has shown that correlation function of two spins cannot be represented by classical correlation of separated bounded random processes. This Bell`s theorem has been interpreted as incompatibility of local realism with quantum mechanics. We show that, in contrast to Bell`s theorem for spin or polarization correlation functions, the correlation of positions (or momenta) of two particles, depending on the rotation angles, in the original EPR model always admits a representation in the form of classical correlation of separated random processes. In this sense there exists a local realistic representation for the original EPR model but there is no such a representation for the Bohm spin reformulation of the EPR paradox. It shows also that the phenomena of quantum nonlocality is based not only on the properties of entangled states but also on the using of particular bounded observables.