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arXiv:quant-ph/0204004 · Physical Review A

Distillable entanglement of multiple copies of Bell states

Yi-Xin Chen, Dong Yang

Published Jul 18, 2002• 12 citations
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Abstract

It is impossible to discriminate four Bell states through local operations and classical communication, if only one copy is provided. To complete this task, two copies will suffice and be necessary. When n copies are provided, we show that the distillable entanglement is exactly $n\ensuremath{-}2.$

DOI
10.1103/physreva.66.014303

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It is impossible to discriminate four Bell states through local operations and classical communication, if only one copy is provided. To complete this task, two copies will suffice and be necessary. When n copies are provided, we show that the distillable entanglement is exactly n\ensuremath−2.n\ensuremath{-}2.n\ensuremath−2.
It is impossible to discriminate four Bell states through local operations and classical communication, if only one copy is provided. To complete this task, two copies will suffice and be necessary. When 2n2n2n copies are provided, we show that the distillable entanglement is exactly $2n\ensuremath{-}2.$ Surprisingly, when $2n\ensuremath{-}1$ copies are provided, the distillable entanglement is also $2n\ensuremath{-}2.$
More than two multipartite orthogonal states cannot always be discriminated if only local operations and classical communication (LOCC) are allowed. We show that four Bell states cannot be discriminated by LOCC, even probabilistically, using the separability properties of a four-party unlockable bound entangled state. Using an existing inequality among the measures of entanglement, we show that any three Bell states cannot be discriminated with certainty by LOCC. Exploiting the inequality, we calculate the distillable entanglement of a certain class of 4 multiply sign in circle 4 mixed states.
We show that the four states $a|00〉+b|11〉, \overline{b}|00〉\ensuremath{-}\ifmmode \bar{a}\else \={a}\fi{}|11〉, c|01〉+d|10〉,$ and $\overline{d}|01〉\ensuremath{-}\overline{c}|10〉$ cannot be discriminated with certainty if only local operations and classical communication (LOCC) are allowed and if only a single copy is provided, except in the case when they are simply ∣00〉,∣11〉,∣01〉,|00〉, |11〉, |01〉,∣00〉,∣11〉,∣01〉, and ∣10〉|10〉∣10〉 (in which case they are trivially distinguishable with LOCC). We go on to show that there exists a continuous range of values of a,b,c,a, b, c,a,b,c, and d such that even three states among the above four are not locally distinguishable, if only a single copy is provided. The proof follows from the fact that logarithmic negativity is an upper bound of distillable entanglement.
We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via local Gaussian operations and classical communication. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [preceding Letter, Phys. Rev. Lett. 89, 137903 ()]] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian states.
There exist sets of multipartite orthogonal states which can not be distinguished by local operation and classical communication (LOCC). The simplest example is the set of four Bell states. The problem of characterizing a set in terms of distinguishability is a hard problem in general. For maximally entangled states, some results are known. Local cloning of entangled states by supplying necessary entanglement in the blank state is another idea that was introduced very recently. Interestingly there are sets of orthogonal entangled states which can be distinguished locally but can not be copied by local operation (even if necessary entanglement is supplied in the blank state) contrary to the fact that happens in the case of global operation.