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arXiv:quant-ph/0310116 · Physical Review A

Separability of quantum states and the violation of Bell-type inequalities

Elena R. Loubenets

Published Apr 5, 2004• 9 citations
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Abstract

In contrast to the widespread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements. In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system. For any separable quantum state, we derive quantum analogs of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended Clauser-Horne-Shimony-Holt inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.

DOI
10.1103/physreva.69.042102

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Comment on “Separability of quantum states and the violation of Bell-type inequalities”

Christoph Simon

The statement by Loubenets [Phys. Rev. A 69, 042102 (2004)] that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell's concept of local hidden variables. In a Bell-type setting, the notion of classicality used by Loubenets [Phys. Rev. A 69, 042102 (2004)] corresponds to the assumption of perfect correlations if the same observable is measured on both sides. While it is obvious that most separable states do not satisfy this assumption, this does not constitute ``nonclassical'' behavior in any usual sense of the word.

arXiv:quant-ph/0410032•Physical Review A•Feb 22, 2005•6 citations
In contrast to the widespread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements. In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system. For any separable quantum state, we derive quantum analogs of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended Clauser-Horne-Shimony-Holt inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.
We introduce a general condition sufficient for the validity of the original Bell inequality (1964) in a local hidden variable (LHV) frame. This condition can be checked experimentally and incorporates only as a particular case the assumption on perfect correlations or anticorrelations usually argued for this inequality in the literature. Specifying this general condition for a quantum bipartite case, we introduce the whole class of bipartite quantum states, separable and nonseparable, that (i) admit an LHV description under any bipartite measurements with two settings per site; (ii) do not necessarily exhibit perfect correlations and may even have a negative correlation function if the same quantum observable is measured at both sites, but (iii) satisfy the ‘perfect correlation’ version of the original Bell inequality for any three bounded quantum observables A 1 , A 2 = B 1 , B 2 at sites ‘ A ’ and ‘ B ’, respectively. Analysing the validity of this general LHV condition under classical and quantum correlation scenarios with the same physical context, we stress that, unlike the Clauser–Horne–Shimony–Holt inequality, the original Bell inequality distinguishes between classicality and quantum separability.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this property does not necessarily exhibit perfect correlations. The class of bipartite states specified by this property includes both separable and nonseparable states. We prove analytically that, for any dimension d ⩾ 3, every Werner state, separable or nonseparable, belongs to this class.
For a nonseparable bipartite quantum state violating the Clauser–Horne–Shimony–Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both 'sides'.
For a nonseparable bipartite quantum state violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both ”sides”. 1
The statement by Loubenets [Phys. Rev. A 69, 042102 (2004)] that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell's concept of local hidden variables. In a Bell-type setting, the notion of classicality used by Loubenets [Phys. Rev. A 69, 042102 (2004)] corresponds to the assumption of perfect correlations if the same observable is measured on both sides. While it is obvious that most separable states do not satisfy this assumption, this does not constitute ``nonclassical'' behavior in any usual sense of the word.