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arXiv:1712.08970

Local distinguishability of quantum states in multipartite System

Xiaoqian Zhang

Published Dec 24, 2017• 6 citations
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Abstract

In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1\otimes m_2\otimes\ldots\otimes m_n$ , where the quantum system $m_1$ belongs to Alice, $m_2$ belongs to Bob, \ldots and $m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1\otimes m_2\otimes m_3$. With the conclusion in $m_1\otimes m_2\otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states in $m_1\otimes m_2\otimes\cdots\otimes m_n$. Our study further reveals quantum nonlocality in multipartite high-dimensional.

DOI
10.48550/arxiv.1712.08970

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In this paper, we give the local distinguishability of four orthogonal product states (OPSs) on multipartite and high-dimension systems. To characterize the different orthogonal relationships between four multipartite OPSs, a new concept, i.e., the vector of the numbers of pairwise orthogonal relations, is proposed. We classify four tripartite OPSs, any two of which are orthogonal only on one subsystem, into three categories by using the vectors of the numbers of pairwise orthogonal relations. We prove that any four tripartite OPSs belonging to two of the three categories can be perfectly distinguished by local operations and classical communication (LOCC). Meanwhile, we find an interesting phenomenon that two sets of four tripartite OPSs belonging to the remaining one of the three categories have different local distinguishability even if they have the same orthogonal graph. This means that the orthogonal graph is not a sufficient condition for determining the local distinguishability of a set of OPSs. Additionally, for the remaining category, we specify the conditions under which four tripartite OPSs can be locally distinguished and the conditions under which four tripartite OPSs cannot be perfectly distinguished by LOCC. On the other hand, we prove that any four tripartite OPSs with seven or more pairwise orthogonal relations can be perfectly distinguished by LOCC. Furthermore, we prove that any four OPSs with six or more pairwise orthogonal relations can be perfectly distinguished by LOCC on four-, five-, and six-partite systems.
We consider the relation between the orthogonality and the distinguishability of a set of arbitrary states (including multipartite states). It is shown that if a set of arbitrary states can be distinguished by local operations and classical communication (LOCC), each of the states can be written as a linear combination of product vectors such that all product vectors of one of the states are orthogonal to the other states. With this result we then prove a simple necessary condition for LOCC distinguishability of a class of orthogonal states. These conclusions may be useful in discussing the distinguishability of orthogonal quantum states further, understanding the essence of nonlocality and discussing the distillation of 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.
Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.
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.
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.