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arXiv:quant-ph/9912002 · Physical Review A

Comment on “Counterfactual entanglement and nonlocal correlations in separable states.”

Daniel R. Terno

Published Dec 4, 2000• 4 citations
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Abstract

The arguments of Cohen [Phys. Rev. A 60, 80 (1999)] against the ``ignorance interpretation'' of mixed states are questioned. The physical arguments are shown to be inconsistent and the supporting example illustrates the opposite of the original statement. The operational difference between two possible definitions of mixed states is exposed and the inadequacy of one of them is stressed.

DOI
10.1103/physreva.63.016101

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The arguments of Cohen [Phys. Rev. A 60, 80 (1999)] against the ``ignorance interpretation'' of mixed states are questioned. The physical arguments are shown to be inconsistent and the supporting example illustrates the opposite of the original statement. The operational difference between two possible definitions of mixed states is exposed and the inadequacy of one of them is stressed.
It is shown that the outcomes of measurements on systems in separable mixed states can be partitioned, via subsequent measurements on a disentangled extraneous system, into subensembles that display the statistics of entangled states. This motivates the introduction of the concept of ``counterfactual'' entanglement, which can be associated with all separable mixed states, including those that are factorable. This type of entanglement gives rise to a kind of postselection-induced Bell inequality violation. The significance of counterfactual entanglement, and its physical implications, are assessed.
In quantum systems, entanglement corresponds to nonclassical correlation of nonlocal observables. Thus, entanglement (or, to the contrary, separability) of a given quantum state is not uniquely determined by properties of the state, but may depend on the choice of the factorization of the algebra of observables. In the present work, we expose and systematize some recently reported results about the possibility to represent a single quantum state as either entangled or separable. We will distinguish in particular the cases of pure and mixed states. For pure states, it has been shown that observables can always be constructed such that any state has any amount of entanglement possible. For mixed states, the situation is more complex and only partial results are known: while it is always possible to choose a factorization such that a state appears separable, a general criterion to determine whether a state can be represented as entangled is not known. These results will be illustrated by several examples, the phenomenon of quantum teleportation, and the geometry of the states of two qubits.
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.