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arXiv:quant-ph/0204055

Bell's Theorem Without Inequalitites for a Maximally Entangled State

LiMing, W.

Published Apr 10, 2002• 0 citations
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Abstract

A proof of Bell's theorem without inequalities for two maximally entangled particles is proposed using the technique of quantum teleportation. It follows Hardy's arguments for a non-maximally entangled state with the help of two auxiliary particles without correlation. The present proof can be tested by measurements with 100% probability.

DOI
10.48550/arxiv.quant-ph/0204055

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A proof of Bell's theorem without inequalities for two maximally entangled particles is proposed using the technique of quantum teleportation. It follows Hardy's arguments for a non-maximally entangled state with the help of two auxiliary particles without correlation. The present proof can be tested by measurements with 100% probability.
We provide a streamlined proof of Hardy's theorem, that almost every entangled state for a pair of quantum particles admits a ``proof of nonlocality'' without inequalities. Moreover, our analysis covers a larger class of observables. Thus we also strengthen Hardy's assertion that the argument fails for maximally entangled states, such as the singlet state. At the same time, we formulate the argument in such a manner that the relations which must be satisfied for local hidden variables are entirely deterministic, making no reference whatsoever to probability, let alone probablistic inequalities.
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The ladder proof of nonlocality without inequalities for two spin-12\frac{1}{2}21​ particles proposed by Hardy [in New Developments on Fundamental Problems in Quantum Physics, edited by M. Ferrero and A. van der Merwe (Kluwer, Dordrecht, 1997)] and Hardy and co-workers [Phys. Rev. Lett. 79, 2755 (1997)] works only for nonmaximally entangled states and goes through for 50% of pairs at the most. A similar ladder proof for two spin-111 particles in a maximally entangled state is presented. In its simplest form, the proof goes through for 17% of pairs. An extended version works for 100% of pairs. The proof can be extended to any maximally entangled state of two spin-sss particles (with s> 1)s>~1)s> 1).
In this paper a new scheme for teleporting an unknown entangled state of two particles is proposed. To weaken the requirement for the quantum channel, without loss of generality, two communicators only share a non-maximally entangled two-particle state. Teleportation can be probabilistically realized if sender performs Bell-state measurements and Hadamard transformation and receiver introduces two auxiliary particles, operates C -not operation, single-qubit measurements and appropriate unitary transformations. The probability of successful teleportation is determined by the smaller one among the coefficients' absolute values of the quantum channel.
As proved by John Bell, quantum mechanics exhibits correlations in spacelike separated bipartite systems that are impossible to reproduce by classical means. There are three levels of "Bell theorems", depending on which aspects of the quantum correlations can or cannot be reproduced classically. The original "Bell inequalities" (Bl) require a perfect classical simulation of all quantum probabilities. With "Bell theorems without inequalities" (BTWI), we ask the classical simulation to be able to produce precisely the outputs that could occur according to quantum mechanics, but we do not worry about their exact probabilities. With "pseudo-telepathy" (PT), we are satisfied if the classical simulation produces only outputs allowed by quantum mechanics, but not necessarily all of them. Bell's original proof of Bl involved a maximally entangled 2times2 bipartite state such as the singlet state. Hardy proved that BTWI are possible in dimension 2times2, but his construction used a non-maximally entangled state. Here, we prove that no 2times2 maximally entangled state can serve to produce BTWI. Combining this with the fact that 2times2 entangled states cannot be used at all for the purpose of PT, it follows a strict hierarchy on the quantum resources that are required to exhibit the various levels of Bell theorems.