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Communications in Theoretical Physics

Locally distinguishing genuinely nonlocal sets with entanglement resource

Qiao Qiao, Su-Juan Zhang, Chen‐Ming Bai, Lu Liu

Published Aug 12, 2024• 2 citations
Publisher

Abstract

Abstract A set of orthogonal product states is deemed genuinely nonlocal if they remain locally indistinguishable under any bipartition. In this paper, we first construct a genuinely nonlocal product basis B I(5, 3) in C 5 ⨂ C 5 ⨂ C 5 using a set of nonlocal product states in C 3 ⨂ C 4 . Then, we obtain a genuinely nonlocal product basis B II(5, 3) by replacing certain states in B I(5, 3) with some superposition states. We achieve perfect discrimination of the constructed genuinely nonlocal product basis, separately employing two EPR states and one GHZ state. Our protocol is more efficient than quantum teleportation.

DOI
10.1088/1572-9494/ad6de6

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Abstract A set of orthogonal product states is deemed genuinely nonlocal if they remain locally indistinguishable under any bipartition. In this paper, we first construct a genuinely nonlocal product basis B I(5, 3) in C 5 ⨂ C 5 ⨂ C 5 using a set of nonlocal product states in C 3 ⨂ C 4 . Then, we obtain a genuinely nonlocal product basis B II(5, 3) by replacing certain states in B I(5, 3) with some superposition states. We achieve perfect discrimination of the constructed genuinely nonlocal product basis, separately employing two EPR states and one GHZ state. Our protocol is more efficient than quantum teleportation.
An orthogonal product basis of a composite Hilbert space is genuinely nonlocal if the basis states are locally indistinguishable across every bipartition. From an operational point of view, such a basis corresponds to a separable measurement that cannot be implemented by local operations and classical communication unless all the parties come together in a single location. In this work we classify genuinely nonlocal product bases into different categories. Our classification is based on the state elimination property of the set via orthogonality-preserving measurements when all the parties are spatially separated or different subsets of the parties come together. We then study local state discrimination protocols for several such bases with additional entangled resources shared among the parties. Apart from consuming less entanglement than teleportation-based schemes, our protocols indicate operational significance of the proposed classification and exhibit nontrivial use of genuine entanglement in the local state discrimination problem.
Genuine nonlocality arises when a set of multipartite orthogonal states is locally indistinguishable under any bipartition of the subsystems. The entanglement-assisted discrimination of such genuinely nonlocal orthogonal product sets has attracted significant attention in quantum information. Based on the criterion of local irreducibility, genuine nonlocality is classified into Type I (reducible) and Type II (irreducible). We present entanglement-assisted discrimination schemes for both types of genuinely nonlocal sets that use minimal resources. For low-dimensional cases, Type I sets require only a single EPR pair, whereas Type II sets necessitate only one GHZ state. We extend these protocols to higher-dimensional systems: the discrimination of Type I sets requires only one maximally entangled state in a two-qutrit system, while that of Type II sets similarly demands a single maximally entangled state in a three-qutrit system. For nnn-partite (n>3n > 3n>3) systems, Type I sets continue to require only one maximally entangled state, whereas Type II sets necessitate just one additional EPR pair compared to their Type I counterparts. These results provide a robust framework for the efficient discrimination of genuinely nonlocal sets using minimal quantum resources.
Entanglement-assisted discrimination of orthogonal quantum states exhibiting quantum nonlocality is a frontier topic in quantum information theory. In this paper, we investigate the role of multipartite entanglement and develop resource-efficient LOCC discrimination protocols for nonlocal sets of orthogonal states, including multipartite orthogonal product-state sets and entangled-state sets with different nonlocal features. By incorporating controlled-NOT (CNOT) operations into the discrimination procedure, we construct protocols for genuinely nonlocal GHZ bases in four- and five-qubit systems that require only a single EPR pair. For the same target sets, we compare different entanglement-assisted schemes and identify those with lower entanglement consumption. We further observe that, on average, protocols avoiding teleportation consume fewer resources than teleportation-based approaches. In addition, when higher-partite GHZ-type resources (with n>3n>3n>3) are available among suitable subsystems, they can in some cases reduce the overall entanglement cost. Our results highlight the operational significance of multipartite entanglement and provide practical protocols for the local discrimination of orthogonal state sets exhibiting quantum nonlocality.
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 .
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 orthogonal product states is said to exhibit "quantum nonlocality without entanglement" if it is locally indistinguishable, i.e. no sequence of local operations and classical communication (LOCC) can perfectly discriminate the states. Building on this foundational idea, recent studies have highlighted the phenomenon of "genuine activation of hidden nonlocality", where a set of initially distinguishable orthogonal states becomes locally indistinguishable through orthogonality-preserving LOCC transformations. In this letter, we establish that any complete orthogonal product basis that is initially locally distinguishable remains so under all orthogonality-preserving local projective measurements, thereby ruling out activation via orthogonality-preserving local projective measurements and classical communication. We further introduce and formalise the notions of "strongly local sets", namely locally distinguishable sets that remain non-activable under all bipartitions. Interestingly, the study of "local activability" of distinguishable sets is useful to characterise the boundary between LOCC distinguishability and its irreversible loss in multipartite systems. Our results provide a rigorous structural understanding of local-to-nonlocal transitions in quantum state discrimination.