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arXiv:quant-ph/0107154 · Physical Review A

Bound entanglement and local realism

Dagomir Kaszlikowski, Marek Żukowski, P. Gnaciński

Published Feb 5, 2002• 9 citations
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Abstract

We show using a numerical approach, which gives necessary and sufficient conditions for the existence of local realism, that the bound entangled state presented in Bennett et al. [Phys. Rev. Lett. 82, 5385 (1999)] admits a local and realistic description. We also find the lowest possible amount of some appropriate entangled state that must be ad-mixed to the bound entangled state so that the resulting density operator has no local and realistic description and as such can be useful in quantum communication and quantum computation.

DOI
10.1103/physreva.65.032107

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We show using a numerical approach, which gives necessary and sufficient conditions for the existence of local realism, that the bound entangled state presented in Bennett et al. [Phys. Rev. Lett. 82, 5385 (1999)] admits a local and realistic description. We also find the lowest possible amount of some appropriate entangled state that must be ad-mixed to the bound entangled state so that the resulting density operator has no local and realistic description and as such can be useful in quantum communication and quantum computation.
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.
Bound entanglement was shown to be activated [P. Horodecki et al., Phys. Rev. Lett. 82, 1056 (1999)] in the sense that the entanglement of a spatially separated two-qutrit system can be increased with nonzero probability via a sufficiently large number of preshared bound-entangled states, local three-level controlled operations, and classical communications. Here we present a local quantum Zeno scheme for activating bound entanglement which is based only on single-particle rotations and threshold measurements. In our scheme, neither a large number of bound-entangled states nor controlled operations are required, and classical communication is required only once at the end of the protocol. We show that a single bound-entangled state is sufficient for increasing the negativity of the target entangled state from 0.11 to 0.17, and by using four more bound-entangled states, negativity can be made greater than 0.42 and the fidelity to the maximally entangled state increases from 0.3 to 0.41, 0.50, 0.59, and 0.61. We believe our results are important not only for quantum technologies but also for a better understanding of quantum entanglement.
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.
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.