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arXiv:2003.06852 · Physical Review A

Nonlocal sets of orthogonal product states in an arbitrary multipartite quantum system

Dong‐Huan Jiang, Guang‐Bao Xu

Published Sep 10, 2020• 51 citations
arXivHTMLPDF

Abstract

Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we first give a simple method to construct a nonlocal set of orthogonal product states in ${\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{d}$, where $d,\phantom{\rule{0.16em}{0ex}}n$ are two positive integers, $d\ensuremath{\ge}2$ and $n\ensuremath{\ge}3$. Then we give a proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer states in the same quantum system. Furthermore, we generalize these two results to a more general ${\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{{d}_{j}}$ quantum system, where $n,\phantom{\rule{0.16em}{0ex}}{d}_{j}$ are two positive integers, $n\ensuremath{\ge}3$ and ${d}_{j}\ensuremath{\ge}2$. Compared with the existing results, the nonlocal set of multipartite orthogonal product states constructed by our method has fewer elements and is simpler. Most of all, our multipartite constructions are more general than the ones reported earlier.

DOI
10.1103/physreva.102.032211

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Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we first give a simple method to construct a nonlocal set of orthogonal product states in \ensuremath⨂j=1nCd{\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{d}\ensuremath⨂j=1n​Cd, where d,nd,\phantom{\rule{0.16em}{0ex}}nd,n are two positive integers, d\ensuremath≥2d\ensuremath{\ge}2d\ensuremath≥2 and n\ensuremath≥3n\ensuremath{\ge}3n\ensuremath≥3. Then we give a proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer states in the same quantum system. Furthermore, we generalize these two results to a more general \ensuremath⨂j=1nCdj{\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{{d}_{j}}\ensuremath⨂j=1n​Cdj​ quantum system, where n,djn,\phantom{\rule{0.16em}{0ex}}{d}_{j}n,dj​ are two positive integers, n\ensuremath≥3n\ensuremath{\ge}3n\ensuremath≥3 and dj\ensuremath≥2{d}_{j}\ensuremath{\ge}2dj​\ensuremath≥2. Compared with the existing results, the nonlocal set of multipartite orthogonal product states constructed by our method has fewer elements and is simpler. Most of all, our multipartite constructions are more general than the ones reported earlier.
Nonlocal sets of orthogonal product states (OPSs) are widely used in quantum protocols owing to their good property. Thus a lot of attention are paid to how to construct a nonlocal set of orthogonal product states though it is a difficult problem. In this paper, we propose a novel general method to construct a nonlocal set of orthogonal product states in Cd⊗Cd\mathbb{C}^{d} \otimes \mathbb{C}^{d}Cd⊗Cd for d≥3d\geq3d≥3. We give an ingenious proof for the local indistinguishability of those product states. The set of product states, which are constructed by our method, has a very good structure. Subsequently, we give a construction of nonlocal set of OPSs with smaller members in Cd⊗Cd\mathbb{C}^{d} \otimes \mathbb{C}^{d}Cd⊗Cd for d≥3d\geq3d≥3. On the other hand, we present two construction methods of nonlocal sets of OPSs in Cm⊗Cn\mathbb{C}^{m} \otimes \mathbb{C}^{n}Cm⊗Cn, where m≥3m\geq3m≥3 and n≥3.n\geq3.n≥3. Furthermore, we propose the concept of isomorphism for two nonlocal sets of OPSs. Our work is of great help to understand the structure and classification of locally indistinguishable OPSs.
The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.
For general bipartite quantum systems, many sets of locally indistinguishable orthogonal product states have been constructed so far. Here, we first present a general method to construct multipartite orthogonal product states in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}4)$ by using some locally indistinguishable bipartite orthogonal product states. And we prove that these multipartite orthogonal quantum states cannot be distinguished by local operations and classical communication. Furthermore, in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}5)$, we give a general method to construct a much smaller number of locally indistinguishable multipartite orthogonal product states for even and odd nnn separately. In addition, we also present a general method to construct complete orthogonal product bases for the multipartite quantum systems. Our results demonstrate the phenomenon of nonlocality without entanglement for the multipartite quantum systems.
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.
Abstract Nonlocal set of orthogonal product states (OPSs) can improve the confidentiality of information when it is used to design quantum cryptographic protocols. It is a difficult question how to construct a nonlocal set of OPSs on general multipartite and high dimensional quantum systems. Different from the previous works, we first present a novel method for constructing a nonlocal product set with 3 d – 2 members on C d ⊗ C d ⊗ C d quantum system for d ≥ 3. Then, we extend this construction method to C d 1 ⊗ C d 2 ⊗ C d 3 quantum system and ⊗ i = 1 n C d i quantum system respectively, where 3 ≤ d 1 ≤ d 2 ≤ d 3 ≤ … ≤ d n and n ≥ 3. The nonlocal set of OPSs constructed by our method contains fewer elements than those constructed by the existing methods, except for one special case. More importantly, the set of states constructed by our method has a completely different structure from those constructed by the existing methods since our nonlocal set does not contain a “stopper” state. Our result is helpful to further understand the different structures of nonlocal sets on multipartite systems.
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.