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arXiv:1502.01274

Detecting the local indistinguishability of maximally entangled states

Sixia Yu, C. H. Oh

Published Feb 4, 2015• 34 citations

Abstract

By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {\it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of $d$ maximally entangled states in a $d\otimes d$ system are systematically constructed for all $d\ge 4$ as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of $k$ locally indistinguishable maximally entangled states with the ratio $k/d$ approaching 3/4. In particular, in a $d\otimes d$ system with even $d\ge 6$, there always exist $d-1$ maximally entangled states that are locally indistinguishable by separable measurements.

DOI: 10.48550/arxiv.1502.01274

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By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {\it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of

d

maximally entangled states in a

d\otimes d

system are systematically constructed for all

d\ge 4

as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of

k

locally indistinguishable maximally entangled states with the ratio

k/d

approaching 3/4. In particular, in a

d\otimes d

system with even

d\ge 6

, there always exist

d-1

maximally entangled states that are locally indistinguishable by separable measurements.

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.

We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3x3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of dxd bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.

We provide a first operational method for checking local indistinguishability of orthogonal states. It originates from that in Ghosh et al. [Phys. Rev. Lett. 87, 5807 (2001)]], though we deal with pure states. Our method shows that probabilistic local distinguishing is possible for a complete multipartite orthogonal basis if and only if all vectors are product. Also, it leads to local indistinguishability of a set of orthogonal pure states of 3 multiply sign in circle 3, which shows that one can have more nonlocality with less entanglement, where "more nonlocality" is in the sense of "increased local indistinguishability of orthogonal states." This is, to our knowledge, the only known example where d orthogonal states in d multiply sign in circle d are locally indistinguishable.

We give an explicit construction of

d

locally indistinguishable, orthogonal maximally entangled states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ for any $d\ensuremath{\ge}4$. This gives an answer to the conjecture proposed by Bandyopadhyay et al. [New J. Phys. 13, 123013 (2011)].

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.