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arXiv:quant-ph/0101084 · Physical Review A

Violations of local realism with quantum systems described byN-dimensional Hilbert spaces up to N=16

Thomas Durt, Dagomir Kaszlikowski, Marek Żukowski

Published Jul 10, 2001• 73 citations

Abstract

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 $N<~9$ were presented. The extension up to $N=16$ 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.

DOI: 10.1103/physreva.64.024101

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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<~9

were presented. The extension up to

N=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.

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.

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)

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.

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 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.