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Scientific Reports

LOCC indistinguishable orthogonal product quantum states

Xiaoqian Zhang, Xiaoqing Tan, Jian Weng, Yongjun Li

Published Jul 5, 2016• 47 citations
Publisher

Abstract

Abstract We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of "Equation missing"2k+i ⊗ "Equation missing"2l+j (i, j ∈ {0, 1} and i ≥ j ) and "Equation missing"3k+i ⊗ "Equation missing"3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of "Equation missing"3k+i ⊗ "Equation missing"3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only "Equation missing"2k ⊗ "Equation missing"2l and "Equation missing"2k+1 ⊗ "Equation missing"2l but also "Equation missing"2k ⊗ "Equation missing"2l+1 and "Equation missing"2k+1 ⊗ "Equation missing"2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of "Equation missing"2k ⊗ "Equation missing"2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.

DOI
10.1038/srep28864

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Abstract We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of "Equation missing"2k+i ⊗ "Equation missing"2l+j (i, j ∈ {0, 1} and i ≥ j ) and "Equation missing"3k+i ⊗ "Equation missing"3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of "Equation missing"3k+i ⊗ "Equation missing"3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only "Equation missing"2k ⊗ "Equation missing"2l and "Equation missing"2k+1 ⊗ "Equation missing"2l but also "Equation missing"2k ⊗ "Equation missing"2l+1 and "Equation missing"2k+1 ⊗ "Equation missing"2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of "Equation missing"2k ⊗ "Equation missing"2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.
In the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.16em}{0ex}}(m,n\ensuremath{\ge}3)$, Y.-L. Wang et al. [Phys. Rev. A 92, 032313 (2015)] presented $3(m+n)\ensuremath{-}9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\ensuremath{\bigotimes}n$. First, in the $3\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3<n)$ quantum system, we construct $3n\ensuremath{-}2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(4\ensuremath{\le}m\ensuremath{\le}n)$, we present $3n+m\ensuremath{-}4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3\ensuremath{\le}m\ensuremath{\le}n)$, we show a smaller set with $2n\ensuremath{-}1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, where ddd is odd, Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)] have constructed d2{d}^{2}d2 orthogonal product basis quantum states that are locally indistinguishable. We find a subset that contains $6d\ensuremath{-}9$ orthogonal product states that are still locally indistinguishable. We generalize our method to an arbitrary bipartite quantum system ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We present a small set with only $3(m+n)\ensuremath{-}9$ orthogonal product states and prove that these states are local operations and classical communication (LOCC) indistinguishable. Even though these $3(m+n)\ensuremath{-}9$ product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication.
In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. In $3\ensuremath{\bigotimes}3$, Bennett et al. [ Phys. Rev. A 59, 1070 (1999)] presented nine orthogonal product basis quantum states which cannot be distinguished by local operations and classical communication (LOCC). In the work by Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)], this result was generalized in $d\ensuremath{\bigotimes}d$, where ddd is odd. In this paper, we aim to construct locally indistinguishable orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. For the general $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$ quantum system, we first construct $4d\ensuremath{-}4$ orthogonal product states, and prove these states are locally indistinguishable using a very simple but quite effective method. Then, based on these states, we construct some classes of locally indistinguishable orthogonal product basis quantum states (OPBS) in $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$. Finally, we construct some LOCC indistinguishable OPBS in multipartite quantum systems. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
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.
For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.
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.