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On a class of small nonlocal set of n-party orthogonal product states

Saronath Halder

Published Mar 28, 2016• 2 citations
OpenAlex

Abstract

It is shown that there exists a class of locally indistinguishable, completable, small set of $n$-party ($n\geq3$) orthogonal product states, with $2n(d-1)$ members in $(d)^{\otimes n}$ dimensions ($d\geq2$), imposing the condition that no party is able to begin with a nontrivial and orthogonality preserving measurement in $d\times d$ dimensions. In fact, it is possible to construct a multipartite nonlocal orthogonal product basis in $(d)^{\otimes n}$ dimensions which contains fewer than $2n(d-1)$ such states (along with other product states) and that does not satisfy the above mentioned condition. As a result of which some information, useful for state discrimination, can be extracted without disturbing the orthogonality of the post measurement states. So, imposing such a condition, is related to the stronger form of nonlocality. Finally, it is shown that a $2\times2$ maximally entangled state as resource, along with a local protocol, is sufficient for perfect discrimination of such a nonlocal set of n-party orthogonal product states in $(d)^{\otimes n}$ dimensions.

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It is shown that there exists a class of locally indistinguishable, completable, small set of nnn-party (n≥3n\geq3n≥3) orthogonal product states, with 2n(d−1)2n(d-1)2n(d−1) members in (d)⊗n(d)^{\otimes n}(d)⊗n dimensions (d≥2d\geq2d≥2), imposing the condition that no party is able to begin with a nontrivial and orthogonality preserving measurement in d×dd\times dd×d dimensions. In fact, it is possible to construct a multipartite nonlocal orthogonal product basis in (d)⊗n(d)^{\otimes n}(d)⊗n dimensions which contains fewer than 2n(d−1)2n(d-1)2n(d−1) such states (along with other product states) and that does not satisfy the above mentioned condition. As a result of which some information, useful for state discrimination, can be extracted without disturbing the orthogonality of the post measurement states. So, imposing such a condition, is related to the stronger form of nonlocality. Finally, it is shown that a 2×22\times22×2 maximally entangled state as resource, along with a local protocol, is sufficient for perfect discrimination of such a nonlocal set of n-party orthogonal product states in (d)⊗n(d)^{\otimes n}(d)⊗n dimensions.
It is known that there exist sets of pure orthogonal product states which cannot be perfectly distinguished by local operations and classical communication (LOCC). Such sets are nonlocal sets which exhibit nonlocality without entanglement. These nonlocal sets can be completable or uncompletable. In this work both completable and uncompletable small nonlocal sets of multipartite orthogonal product states are constructed. Apart from nonlocality, these sets have other interesting properties. In particular, the completable sets lead to the construction of a class of complete orthogonal product bases with the property that if such a basis is given then no state can be eliminated from that basis by performing orthogonality-preserving measurements. On the other hand, an uncompletable set of the present kind contains several Shifts unextendible product bases (UPBs) that belong to qubit subspaces. Identifying these subspace UPBs, it is possible to obtain a class of high-dimensional multipartite bound entangled states. Finally, it is shown that a two-qubit maximally entangled Bell state shared between any two parties is sufficient as a resource to distinguish the states of any completable set (of the above kind) perfectly by LOCC. This constitutes an example where the amount of entanglement, sufficient to accomplish the aforesaid task, depends neither on the dimension of the individual subsystems nor on the number of parties.
Nonlocality without entanglement and its subsequent generalizations offer deep information-theoretic insights and subsequently find several useful applications. The concept of a genuinely nonlocal set of product states emerges as a natural multipartite generalization of this phenomenon. The existence of such sets eventually raises the problem concerning their entanglement-assisted discrimination. Here, we construct examples of genuinely nonlocal product states for an arbitrary number of parties. The strength of genuine nonlocality of these sets can be considered minimal as their perfect discrimination is possible with entangled resources residing in Hilbert spaces having the smallest possible dimensions. Our constructions lead to fully separable measurements that are impossible to implement even if all but one party come together. Furthermore, they also provide the opportunity to compare different multipartite states that otherwise are incomparable under single copy local manipulation.
In multipartite systems, great progress has been made recently on the study of strong quantum nonlocality without entanglement. However, the existence of orthogonal product sets with strong quantum nonlocality in even party systems remains unknown. Here the even number is greater than 4. In this paper, we successfully construct strongly nonlocal orthogonal product sets in nnn-partite systems for all even nnn, which answers the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)] for any possible even party systems. Thus, we find general construction of strongly nonlocal orthogonal product sets in space ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathcal{C}}^{{d}_{i}}$ $(n,{d}_{i}\ensuremath{\ge}3)$ and show that there do exist incomplete orthogonal product bases that can be strongly nonlocal in any possible nnn-partite systems for all even nnn. Our newly constructed orthogonal product sets are asymmetric. We analyze the differences and connections between these sets and the known orthogonal product sets in odd party systems. In addition, we present a local state discrimination protocol for our sets by using additional entangled resources. When at least two subsystems have dimensions greater than three, the protocol consumes less entanglement than a teleportation-based protocol. A strongly nonlocal set implies that the information cannot be completely accessed as long as it does not happen that all parties are together. As an application, we connect our sets with local information hiding in multipartite systems.
Local distinguishability of orthogonal quantum states is an area of active research in quantum information theory. However, most of the relevant results are about local distinguishability in bipartite Hilbert space and very little is known about the multipartite case. In this paper we present a generic method to construct a completable nnn-partite $(n\ensuremath{\ge}3)$ product basis with only 2n2n2n members, which exhibits nonlocality without entanglement with nnn parties, each holding a system of any finite dimension. We give an effective proof of the nonlocality of the completable multipartite product basis. In addition, we construct another incomplete multipartite product basis with a smaller number of members that cannot be distinguished by local operations and classical communication in a ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ quantum system, where $n\ensuremath{\ge}3$ and ${d}_{i}\ensuremath{\ge}2$ for i=1,2,...,ni=1,2,...,ni=1,2,...,n. The results can lead to a better understanding of the phenomenon of nonlocality without entanglement in any multipartite quantum system.
We study the constructions of nonlocal orthogonal product states in multipartite systems that cannot be distinguished by local operations and classical communication. We first present two constructions of nonlocal orthogonal product states in tripartite systems Cd⊗Cd⊗Cd (d≥3)\mathcal{C}^{d}\otimes\mathcal{C}^{d}\otimes\mathcal{C}^{d}~(d\geq3)Cd⊗Cd⊗Cd (d≥3) and Cd⊗Cd+1⊗Cd+2 (d≥3)\mathcal{C}^d\otimes \mathcal{C}^{d+1}\otimes \mathcal{C}^{d+2}~(d\geq 3)Cd⊗Cd+1⊗Cd+2 (d≥3). Then for general tripartite quantum system Cn1⊗Cn2⊗Cn3\mathcal{C}^{n_{1}}\otimes\mathcal{C}^{n_{2}}\otimes\mathcal{C}^{n_{3}}Cn1​⊗Cn2​⊗Cn3​ (3≤n1≤n2≤n3)(3\leq n_{1}\leq n_{2}\leq n_{3})(3≤n1​≤n2​≤n3​), we obtain 2(n2+n3−1)−n12(n_{2}+n_{3}-1)-n_{1}2(n2​+n3​−1)−n1​ nonlocal orthogonal product states. Finally, we put forward a new construction approach in Cd1⊗Cd2⊗⋯⊗Cdn\mathcal{C}^{d_{1}}\otimes \mathcal{C}^{d_{2}}\otimes\cdots\otimes \mathcal{C}^{d_{n}}Cd1​⊗Cd2​⊗⋯⊗Cdn​ (d1,d2,⋯dn≥3, n>6)(d_1,d_2,\cdots d_n\geq3,\, n>6)(d1​,d2​,⋯dn​≥3,n>6) multipartite systems. Remarkably, our indistinguishable sets contain less nonlocal product states than the existing ones, which improves the recent results and highlights their related applications in quantum information processing.
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.