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arXiv:quant-ph/0106156 · Journal of Physics A Mathematical and General

Violations of local realism by two entangled quNits

Daniel Collins, Sandu Popescu

Published Aug 24, 2001• 2 citations
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Abstract

Results obtained in two recent papers, (Kaszlikowski D, Gnacinski P, Zukowski M, Miklaszewski W and Zeilinger A 2000 Phys. Rev. Lett. 85 4418 and Durt T, Kaszlikowski D and Zukowski M 2001 Preprint quant-ph/0101084) seem to indicate that the nonlocal character of the correlations between the outcomes of measurements performed on entangled systems separated in space is not robust in the presence of noise. This is surprising, since entanglement itself is robust. Here we revisit this problem and argue that the class of gedanken experiments considered in the two recent papers listed above is too restrictive. By considering a more general class, involving sequences of measurements, we prove that the nonlocal correlations are in fact robust.

DOI
10.1088/0305-4470/34/35/309

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Results obtained in two recent papers, (Kaszlikowski D, Gnacinski P, Zukowski M, Miklaszewski W and Zeilinger A 2000 Phys. Rev. Lett. 85 4418 and Durt T, Kaszlikowski D and Zukowski M 2001 Preprint quant-ph/0101084) seem to indicate that the nonlocal character of the correlations between the outcomes of measurements performed on entangled systems separated in space is not robust in the presence of noise. This is surprising, since entanglement itself is robust. Here we revisit this problem and argue that the class of gedanken experiments considered in the two recent papers listed above is too restrictive. By considering a more general class, involving sequences of measurements, we prove that the nonlocal correlations are in fact robust.
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.
Predictions for systems in entangled states cannot be described in local realistic terms. However, after admixing some noise such a description is possible. We show that for two quantum systems described by N-dimensional Hilbert spaces (quNits) in a maximally entangled state the minimal admixture of noise increases monotonically with N. The results are a direct extension of those of Kaszlikowski et al. [Phys. Rev. Lett. 85, 4418 (2000)], where results for N< 9N<~9N< 9 were presented. The extension up to N=16N=16N=16 is possible when one defines for each N a specially chosen set of observables. We also present results concerning the critical detectors efficiency beyond which a valid test of local realism for entangled quNits is possible.
We report an exhaustive numerical analysis of violations of local realism by two qutrits in all possible pure entangled states. In Bell-type experiments we allow any pair of local unitary U(3) transformations to define the measurement bases. Surprisingly, Schmidt-rank-2 states, resembling pairs of maximally entangled qubits, lead to the most noise-robust violations of local realism. The phenomenon seems to be even more pronounced for four- and five-dimensional systems, for which we tested a few interesting examples.
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.
Tests of local realism versus quantum mechanics based on Bell's inequality employ two entangled qubits. We investigate the general case of two entangled quantum systems defined in N-dimensional Hilbert spaces, or " quNits." Via a numerical linear optimization method we show that violations of local realism are stronger for two maximally entangled quNits ( 3</=N</=9) than for two qubits and that they increase with N. The two quNit measurements can be experimentally realized using entangled photons and unbiased multiport beam splitters.
Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.