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arXiv:1310.4220 · Physical Review A

Three maximally entangled states can require two-way local operations and classical communication for local discrimination

Michael Nathanson

Published Dec 12, 2013• 69 citations
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Abstract

We show that there exist sets of three mutually orthogonal $d$-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of $d$. 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 $d$-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.

DOI
10.1103/physreva.88.062316

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One-way local operations and classical communication (LOCC) can fail to distinguish four orthogonal maximally entangled states in every local dimension d≥4d\ge4d≥4. We show that every set of kkk pairwise orthogonal maximally entangled states in Cd⊗Cd\mathbb C^d\otimes\mathbb C^dCd⊗Cd is perfectly distinguishable by an A→B→AA\to B\to AA→B→A protocol whenever d≥Ck4[log⁡(2k)]νd\ge Ck^4[\log(2k)]^νd≥Ck4[log(2k)]ν, for universal constants C,ν>0C,ν>0C,ν>0. A single feedback message therefore raises the number of candidates that can always be distinguished from a constant to Ω ⁣(d1/4/(log⁡d)ν/4)Ω\!\left(d^{1/4}/(\log d)^{ν/4}\right)Ω(d1/4/(logd)ν/4), establishing a polynomial separation between one-way and two-way LOCC. In our protocol, Alice preserves exact orthogonality while reducing overlaps between Bob's conditional subspaces. Bob reduces the candidates to a singleton or an orthogonal pair and refines his outcome so that Alice can complete the discrimination. As a corollary, nine copies suffice for perfect two-way LOCC discrimination of any orthogonal maximally entangled set in all sufficiently large local dimensions.
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.
Two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C3⊗C3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in Cn⊗Cm. In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.
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.
In this work, we construct small sets of bipartite orthogonal pure states that cannot be perfectly distinguished by local operations and classical communication (LOCC). We mention that not all the states within the constructed sets are necessarily entangled. However, such a set contains at least one entangled state which cannot be conclusively identified by LOCC (with nonzero probability). Then, we show that the states of any such set can be perfectly distinguished by LOCC using a minimal-dimensional entangled resource state. Clearly, here the entangled resource state provides an advantage irrespective of the dimension of the given set. Using this result, we also prove that any pure entangled state is useful as a resource to distinguish the states of any present set unambiguously with nonzero probability under LOCC. These sets exist in all two-qudit Hilbert spaces. Furthermore, it is possible to decrease the average entanglement content of such a set or to increase the cardinality of the set without changing certain properties of the same.
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.