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Physical Review A

Local discrimination of four or more maximally entangled states

Guojing Tian, Sixia Yu, Fei Gao, Qiaoyan Wen, C. H. Oh

Published May 15, 2015• 33 citations
Publisher

Abstract

In the $4\ensuremath{\bigotimes}4$ quantum system, four qubit lattice states are found to be either locally indistinguishable or distinguishable by using one-way classical communication only, presenting a curious and counterintuitive case in which two-way classical communication has no advantage. Based on these results, we explicitly construct sets of four, five, and even more maximally entangled states that can be locally distinguished only with the help of two-way classical communication, solving partially the open problem described by Nathanson [Phys. Rev. A 88, 062316 (2013)] and highlighting in more detail the difference between local operations with and without two-way classical communication in distinguishing more than three states.

DOI
10.1103/physreva.91.052314

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In the 4\ensuremath⨂44\ensuremath{\bigotimes}44\ensuremath⨂4 quantum system, four qubit lattice states are found to be either locally indistinguishable or distinguishable by using one-way classical communication only, presenting a curious and counterintuitive case in which two-way classical communication has no advantage. Based on these results, we explicitly construct sets of four, five, and even more maximally entangled states that can be locally distinguished only with the help of two-way classical communication, solving partially the open problem described by Nathanson [Phys. Rev. A 88, 062316 (2013)] and highlighting in more detail the difference between local operations with and without two-way classical communication in distinguishing more than three states.
We present a necessary condition for two sets of maximally entangled states to be locally equivalent in terms of local invariants. With the help of these invariants, first, we are able to completely classify all the distinguishable and indistinguishable sets of four generalized Bell states in the $4\ensuremath{\bigotimes}4$ quantum system: there are ten inequivalent classes including three locally indistinguishable and seven locally distinguishable quadruples. All distinguishable quadruples in this case can be distinguished by using only one-way classical communication, implying that the two-way classical communication has no advantage over one-way classical communication, similar to the case of qubit lattice states. Second, we are able to completely classify all the triplets of generalized Bell states in the quantum systems of all prime dimensions as well as in some composite dimensions.
We explicitly exhibit a set of four ququad-ququad orthogonal maximally entangled states that cannot be perfectly distinguished by means of local operations and classical communication. Before our work, it was unknown whether there is a set of d locally indistinguishable d⊗d orthogonal maximally entangled states for some positive integer d. We further show that a 2⊗2 maximally entangled state can be used to locally distinguish this set of states without being consumed, thus demonstrate a novel phenomenon of entanglement discrimination catalysis. Based on this set of states, we construct a new set K consisting of four locally indistinguishable states such that K(⊗m) (with 4(m) members) is locally distinguishable for some m greater than one. As an immediate application, we construct a noisy quantum channel with one sender and two receivers whose local zero-error classical capacity can achieve the full dimension of the input space but only with a multi-shot protocol.
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.
A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs) for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We also construct several four-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one of which is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for four-GBS sets in the quantum system ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$, the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of four-GBS sets in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$ is completely resolved.
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.
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.