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arXiv:quant-ph/0111136 · Physical Review A

Local indistinguishability of orthogonal pure states by using a bound on distillable entanglement

Sibasish Ghosh A, Guruprasad Kar, Anirban Roy, Debasis Sarkar, Aditi Sen, Ujjwal Sen

Published Jun 4, 2002• 43 citations
arXivHTMLPDF

Abstract

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 $|00〉, |11〉, |01〉,$ and $|10〉$ (in which case they are trivially distinguishable with LOCC). We go on to show that there exists a continuous range of values of $a, b, c,$ 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.

DOI
10.1103/physreva.65.062307

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We show that the four states a∣00〉+b∣11〉,b‾∣00〉\ensuremath−\ifmmodeaˉ\elseaˉ\fi∣11〉,c∣01〉+d∣10〉,a|00〉+b|11〉, \overline{b}|00〉\ensuremath{-}\ifmmode \bar{a}\else \={a}\fi{}|11〉, c|01〉+d|10〉,a∣00〉+b∣11〉,b∣00〉\ensuremath−\ifmmodeaˉ\elseaˉ\fi∣11〉,c∣01〉+d∣10〉, and d‾∣01〉\ensuremath−c‾∣10〉\overline{d}|01〉\ensuremath{-}\overline{c}|10〉d∣01〉\ensuremath−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.
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 present a complete characterization for the local distinguishability of orthogonal $2\ensuremath{\bigotimes}3$ pure states except for some special cases of three states. Interestingly, we find there is a large class of four or three states that are indistinguishable by local projective measurements and classical communication (LPCC), but can be perfectly distinguished by LOCC. That indicates the ability of LOCC for discriminating $2\ensuremath{\bigotimes}3$ states is strictly more powerful than that of LPCC, which is strikingly different from the case of multiqubit states. We also show that classical communication plays a crucial role for local distinguishability by constructing a class of $m\ensuremath{\bigotimes}n$ states which require at least $2\phantom{\rule{0.2em}{0ex}}\mathrm{min}{m,n}\ensuremath{-}2$ rounds of classical communication in order to achieve a perfect local discrimination.
Abstract It is an interesting topic how much entanglement resources are necessary to distinguish locally indistinguishable orthogonal quantum states by local operations and classical communication (LOCC). Using only one ebit of entanglement some locally indistinguishable orthogonal product bases can be distinguished by LOCC [2019 Phys. Rev. A 99 012343]. However, it is still an open question whether one ebit of entanglement is sufficient for distinguishing a locally indistinguishable set of orthogonal maximally entangled states. We focus on the generalized Bell states (GBSs), and firstly find that any four GBSs in a 4 ⊗ 4 quantum system are always distinguished by one-way LOCC using one ebit of entanglement. Then we consider a general case, i.e., d ⊗ d quantum system ( d is even), and show that the set L of any GBSs can be distinguished by one-way LOCC using one ebit of entanglement if S A \ S L is nonempty, where S A is the admissible solutions set of L and S L is the genuinely nonadmissible solutions set of L . This criterion is very simple and convenient since only some congruence equations are calculated. Finally, we find a curious and counterintuitive case, i.e., in a 3 ⊗ 3 quantum system there exist four GBSs which cannot be distinguished by one-way LOCC using one ebit of entanglement, although the dimension of the subsystem is less than four. These results can help us better understand the role of entanglement as a resource in local state discrimination problems.
In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than ddd numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any ddd or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).
In this paper, we mainly study the local indistinguishability of mutually orthogonal bipartite maximally entangled states. We construct sets of fewer than ddd orthogonal maximally entangled states which are not distinguished by one-way local operations and classical communication (LOCC) in the Hilbert space of $d\ensuremath{\bigotimes}d$. The proof, based on the Fourier transform of an additive group, is very simple but quite effective. Simultaneously, our results give a general unified upper bound for the minimum number of one-way LOCC indistinguishable maximally entangled states. This improves previous results which only showed sets of $N\ensuremath{\ge}d\ensuremath{-}2$ such states. Finally, our results also show that previous conjectures in Zhang et al. [Z.-C. Zhang, Q.-Y. Wen, F. Gao, G.-J. Tian, and T.-Q. Cao, Quant. Info. Proc. 13, 795 (2014)] are indeed correct.
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.