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Laser Physics Letters

Strong nonlocality of orthogonal entangled states and entanglement-assisted discrimination

Jia-Xin Ma, Chen‐Ming Bai, Su-Juan Zhang

Published Dec 1, 2025• 1 citation
Publisher

Abstract

Abstract In this paper, we construct strongly nonlocal orthogonal entangled sets (SNOESs) in C 4 ⊗ C 3 ⊗ C 5 and C 5 ⊗ C 4 ⊗ C 6 . To better understand SNOESs, we introduce the concept of unitary equivalence for such sets, defined via transformations under unitary matrices. For C d 1 ⊗ C d 2 ⊗ C d 3 , we show that applying arbitrary unitary operators to a known SNOES creates new SNOESs, and we demonstrate this explicitly for C 4 ⊗ C 3 ⊗ C 5 . Our results show that any orthogonal entangled set (OES) derived from an SNOES is locally irreducible under all bipartitions, thus establishing a universal criterion for the preservation of strong nonlocality under all unitary transformations. Most importantly, the proposed method for identifying equivalent SNOESs is concise and widely applicable. Furthermore, we propose an entanglement-assisted protocol for local discrimination of our results. The protocol significantly optimizes entangled resource consumption, highlighting the practical relevance of our framework for quantum information tasks reliant on strongly nonlocal state discrimination. Our results, which can be used to classify strongly nonlocal OESs in tripartite systems, advance the theory of quantum information and deepen the structural understanding of strongly nonlocal sets.

DOI
10.1088/1612-202x/ae27a4

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Abstract In this paper, we construct strongly nonlocal orthogonal entangled sets (SNOESs) in C 4 ⊗ C 3 ⊗ C 5 and C 5 ⊗ C 4 ⊗ C 6 . To better understand SNOESs, we introduce the concept of unitary equivalence for such sets, defined via transformations under unitary matrices. For C d 1 ⊗ C d 2 ⊗ C d 3 , we show that applying arbitrary unitary operators to a known SNOES creates new SNOESs, and we demonstrate this explicitly for C 4 ⊗ C 3 ⊗ C 5 . Our results show that any orthogonal entangled set (OES) derived from an SNOES is locally irreducible under all bipartitions, thus establishing a universal criterion for the preservation of strong nonlocality under all unitary transformations. Most importantly, the proposed method for identifying equivalent SNOESs is concise and widely applicable. Furthermore, we propose an entanglement-assisted protocol for local discrimination of our results. The protocol significantly optimizes entangled resource consumption, highlighting the practical relevance of our framework for quantum information tasks reliant on strongly nonlocal state discrimination. Our results, which can be used to classify strongly nonlocal OESs in tripartite systems, advance the theory of quantum information and deepen the structural understanding of strongly nonlocal sets.
Strong quantum nonlocality was introduced recently as a stronger manifestation of nonlocality in multipartite systems through the notion of local irreducibility in all bipartitions. Known existing results for sets of strongly nonlocal orthogonal states are limited to product states. In this paper, based on the Rubik's cube, we give a construction of such sets consisting of entangled states in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$ for all $d\ensuremath{\ge}3$. Consequently, we answer an open problem given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], that is, orthogonal entangled bases that are strongly nonlocal do exist. Furthermore, we propose two entanglement-assisted protocols for local discrimination of our results. Each protocol consumes less entanglement resources than the teleportation-based protocol on average. Our results exhibit the phenomenon of strong quantum nonlocality with entanglement.
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.
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.
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.
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.
In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality in multipartite quantum systems. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving positive operator-valued measures on fixed subsystem and partially answer an open question given by Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. The connection between the nonlocality and the plane structure of OPSs is established. We successfully construct a strongly nonlocal OPS in ${\mathcal{C}}^{{d}_{A}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{B}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{C}}\phantom{\rule{4pt}{0ex}}({d}_{A},{d}_{B},{d}_{C}\ensuremath{\ge}4)$, which contains fewer quantum states, and generalize the structures of known OPSs to any possible three and four-partite systems. In addition, we propose several entanglement-assisted protocols for perfectly local discrimination of the sets. It is shown that the protocols without teleportation use less entanglement resources that on average and these sets can always be discriminated locally with multiple copies of two-qubit maximally entangled states. These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.