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arXiv:quant-ph/9908039

Quantum perfect correlations and Hardy's nonlocality theorem

Jose Luis Cereceda

Published Aug 12, 1999• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/9908039

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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.
Hardy’s non-locality provides a proof of the incompatibility between quantum mechanics and local realism without using Bell inequalities. While this argument has been extensively studied for two- and three-qubit systems, a detailed analysis of the four-qubit case is still lacking. In this work, we investigate Hardy’s non-locality for a four-qubit system within the standard two-setting framework. We explicitly construct the entangled state satisfying the Hardy conditions and determine the measurement settings that maximize the success probability. • We present an explicit analytical construction of a four-qubit Hardy state together with the corresponding projective measurement settings. • The maximum Hardy success probability for four qubits is found to be 0.143, exceeding the values obtained in the two- and three-qubit cases. • The resulting state is non-maximally entangled and exhibits genuine multipartite quantum correlations.
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.
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.
The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell's inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations.
Nonlocality without inequality is an elegant argument introduced by Hardy for two qubit systems, and later generalised to n qubits, to establish contradiction of quantum theory with local realism. Interestingly, for n = 2 this argument is actually a corollary of Bell-type inequalities, viz., the CH–Hardy inequality involving Bell correlations, but for n greater than two it involves n-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for n qubits and then associated with each such inequality, we provide a new Hardy-type local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'son-type theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.
We generalize Hardy's proof of nonlocality to the case of bipartite mixed statistical operators, and we exhibit a necessary condition which has to be satisfied by any given mixed state $\ensuremath{\sigma}$ in order that a local and realistic hidden variable model exists which accounts for the quantum mechanical predictions implied by $\ensuremath{\sigma}$. Failure of this condition will imply both the impossibility of any local explanation of certain joint probability distributions in terms of hidden variables and the nonseparability of the considered mixed statistical operator. Our result can be also used to determine the maximum amount of noise, arising from imperfect experimental implementations of the original Hardy's proof of nonlocality, in presence of which it is still possible to put into evidence the nonlocal features of certain mixed states.