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Physical Review A

sComment on 'Local-realism violations in two-particle interferometry'

Jose Luis Cereceda

Published May 1, 1997• 2 citations
Publisher

Abstract

In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found to fail whenever the dimensionality of the product Hilbert space is reduced to four. The explanation for this failure is provided by the fact according to which no direct contradiction between the quantum-mechanical and locally realistic predictions can arise for any fraction of an ensemble of 2\ifmmode\times\else\texttimes\fi{}2 systems configured in the maximally entangled state.

DOI
10.1103/physreva.55.3968

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In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found to fail whenever the dimensionality of the product Hilbert space is reduced to four. The explanation for this failure is provided by the fact according to which no direct contradiction between the quantum-mechanical and locally realistic predictions can arise for any fraction of an ensemble of 2\ifmmode\times\else\texttimes\fi{}2 systems configured in the maximally entangled state.
In Kaszlikowski et al. [Phys. Rev. Lett. 85, 4418 (2000)], it has been shown numerically that the violation of local realism for two maximally entangled N-dimensional (3< N)(3<~N)(3< N) quantum objects is stronger than for two maximally entangled qubits and grows with N. In this paper, we present the analytical proof of this fact for N=3.N=3.N=3.
In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions $e{\ensuremath{\rightarrow}}_{1},$ $e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},$ $e{\ensuremath{\rightarrow}}_{2},$ $e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},$ and $e{\ensuremath{\rightarrow}}_{3}.$ It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states $|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.$ It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.
In this paper the failure of Hardy's nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D_11,D_21), (D_11,D_22) and (D_12,D_21) [in the sense of having with certainty equal (different) readings for a joint measurement of any one of the pairs (D_11,D_21), (D_11,D_22), and (D_12,D_21)], necessarily entails perfect correlation for the pair of observables (D_12,D_22) [in the sense of having with certainty equal (different) readings for a joint measurement of the pair (D_12,D_22)]. Indeed, the set of these four perfect correlations is found to satisfy the CHSH inequality, and then no violations of local realism will arise for the maximally entangled state as far as the four observables D_ij, i,j = 1,2, are concerned. The connection between this fact and the impossibility for the quantum mechanical predictions to give the maximum possible theoretical violation of the CHSH inequality is pointed out. Moreover, it is generally proved that the fulfillment of all the Hardy nonlocality conditions necessarily entails a violation of the resulting CHSH inequality. The largest violation of this latter inequality is determined.
In this paper we extend the ladder proof of nonlocality without inequalities for two spin-half particles given by Boschi et al (Boschi D et al 1997 Phys. Rev. Lett. 79 2755) to the case in which the measurement settings of the apparatus measuring one of the particles are different from the measurement settings of the apparatus measuring the other particle. It is shown that, in any case, the proportion of particle pairs for which the contradiction with local realism goes through is maximized when the measurement settings are the same for each apparatus. Also we write down a Bell inequality for the experiment in question which is violated by quantum mechanics by an amount which is twice as much as the amount by which quantum mechanics violates the Bell inequality considered in the above paper by Boschi et al .