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Physical Review A

Distinguishing maximally entangled states by one-way local operations and classical communication

Zhichao Zhang, Keqin Feng, Fei Gao, Qiaoyan Wen

Published Jan 21, 2015• 26 citations
Publisher

Abstract

In this paper, we mainly study the local indistinguishability of mutually orthogonal bipartite maximally entangled states. We construct sets of fewer than $d$ 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.

DOI
10.1103/physreva.91.012329

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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⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂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≥d\ensuremath−2N\ensuremath{\ge}d\ensuremath{-}2N\ensuremath≥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.
We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k≤dk \leq dk≤d orthogonal maximally entangled states in $\complex^{d}\otimes\complex^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case k=dk = dk=d. We give, for the first time, a proof that such sets of states indeed exist even in the case k<dk < dk<d. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.
We consider the question of perfect local distinguishability of mutually orthogonal bipartite quantum states, with the property that every state can be specified by a unitary operator acting on the local Hilbert space of Bob. We show that if the states can be exactly discriminated by one-way local operations and classical communication (LOCC) where Alice goes first, then the unitary operators can also be perfectly distinguished by an orthogonal measurement on Bob's Hilbert space. We give examples of sets of N ⩽ d maximally entangled states in d ⊗ d for d = 4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly, for d = 5,6, our examples consist of four and five states, respectively. We conjecture that these states cannot be perfectly discriminated by two-way LOCC.
For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.
We show that there exist sets of three mutually orthogonal ddd-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of ddd. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal ddd-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
In the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.16em}{0ex}}(m,n\ensuremath{\ge}3)$, Y.-L. Wang et al. [Phys. Rev. A 92, 032313 (2015)] presented $3(m+n)\ensuremath{-}9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\ensuremath{\bigotimes}n$. First, in the $3\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3<n)$ quantum system, we construct $3n\ensuremath{-}2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(4\ensuremath{\le}m\ensuremath{\le}n)$, we present $3n+m\ensuremath{-}4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3\ensuremath{\le}m\ensuremath{\le}n)$, we show a smaller set with $2n\ensuremath{-}1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than ddd which are not distinguished by one-way local operations and classical communication (1-LOCC) in d⊗dd\otimes dd⊗d. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang \emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang \emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that ddd is odd of the results also shows that the set of 4 GBSs in 5⊗55\otimes 55⊗5 in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fan's method.