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arXiv:2011.10286 · Journal of Physics A Mathematical and Theoretical

Local distinguishability based genuinely quantum nonlocality without entanglement

Mao-Sheng Li, Yan-Ling Wang, Fei Shi, Man-Hong Yung

Published Sep 21, 2021• 26 citations
arXivHTMLPDF

Abstract

Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .

DOI
10.1088/1751-8121/ac28cd

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Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .
Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.
A set of orthogonal quantum states in multipartite systems is of genuine nonlocality if it is locally indistinguishable in every bipartition. If it is locally reducible when the parties are separated, we say that it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 1}; otherwise, it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 2}. For a locally distinguishable set without local redundancy, if there exist some orthogonality preserving local measurements such that each outcome leads to a locally indistinguishable set, then we say that it exhibits the activation of nonlocality. We activate type-\uppercase\expandafter{\romannumeral 1} and type-\uppercase\expandafter{\romannumeral 2} genuine nonlocality of orthogonal product state sets in tripartite systems. In particular, we tackle the local irredundancy problem with partial trace operation and ppp-ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.
Strong nonlocality based on local distinguishability is a stronger form of quantum nonlocality recently introduced in multipartite quantum systems: an orthogonal set of multipartite quantum states is said to have the property of strong nonlocality if it is locally irreducible for every bipartition of the subsystems. Most of the known results are limited to sets with product states. Shi et al. presented the first result of strongly nonlocal sets with entanglement [Phys. Rev. A 102, 042202 (2020)], and they questioned the existence of a strongly nonlocal set with genuine entanglement. In this work, we relate the strong nonlocality of some special set of genuine entanglement to the connectivities of some graphs. Using this relation, we successfully construct sets of genuinely entangled states with strong nonlocality. As a consequence, our constructions give a negative answer to Shi et al.'s question and also provide an answer to the open problem raised by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. This work associates a physical quantity named strong nonlocality with a mathematical quantity called graph connectivity.
A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general nnn-partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d for any d≥3d\geq 3d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.