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arXiv:1411.6702 · Physical Review A

d locally indistinguishable maximally entangled states in Cd⊗Cd

Mao-Sheng Li, Yan-Ling Wang, Shao-Ming Fei, Zhu‐Jun Zheng

Published Apr 14, 2015• 25 citations
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Abstract

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)].

DOI
10.1103/physreva.91.042318

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We give an explicit construction of ddd locally indistinguishable, orthogonal maximally entangled states in Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd for any d\ensuremath≥4d\ensuremath{\ge}4d\ensuremath≥4. This gives an answer to the conjecture proposed by Bandyopadhyay et al. [New J. Phys. 13, 123013 (2011)].
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 ddd maximally entangled states in a d⊗dd\otimes dd⊗d system are systematically constructed for all d≥4d\ge 4d≥4 as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of kkk locally indistinguishable maximally entangled states with the ratio k/dk/dk/d approaching 3/4. In particular, in a d⊗dd\otimes dd⊗d system with even d≥6d\ge 6d≥6, there always exist d−1d-1d−1 maximally entangled states that are locally indistinguishable by separable measurements.
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.
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.
In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. In $3\ensuremath{\bigotimes}3$, Bennett et al. [ Phys. Rev. A 59, 1070 (1999)] presented nine orthogonal product basis quantum states which cannot be distinguished by local operations and classical communication (LOCC). In the work by Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)], this result was generalized in $d\ensuremath{\bigotimes}d$, where ddd is odd. In this paper, we aim to construct locally indistinguishable orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. For the general $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$ quantum system, we first construct $4d\ensuremath{-}4$ orthogonal product states, and prove these states are locally indistinguishable using a very simple but quite effective method. Then, based on these states, we construct some classes of locally indistinguishable orthogonal product basis quantum states (OPBS) in $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$. Finally, we construct some LOCC indistinguishable OPBS in multipartite quantum systems. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
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 solve the unextendible maximally entangled basis (UMEB) problem in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}(d\ensuremath{\ne}{d}^{\ensuremath{'}}).$ The results show that a UMEB will always exist. In addition, there might be two sets of UMEBs with different numbers. The main difficulty is to prove the unextendibility of the set of states. We give an explicit construction of UMEBs by considering the Schmidt number of the complementary space of the states we construct.