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arXiv:1905.11121 · Physical Review A

Distinguishability classes, resource sharing, and bound entanglement distribution

Saronath Halder, Ritabrata Sengupta

Published Jan 10, 2020• 25 citations
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Abstract

Suppose a set of $m$-partite, $m\ensuremath{\ge}3$, pure orthogonal fully separable states is given. We consider the task of distinguishing these states perfectly by local operations and classical communication (LOCC) in different $k$ partitions, $1<k<m$. Based on this task, it is possible to classify the sets of product states into different classes. For tripartite systems, a classification of the sets with explicit examples is presented. Few important cases related to the aforesaid task are also studied when the number of parties, $m\ensuremath{\ge}4$. These cases never appear for a tripartite system. However, to distinguish any LOCC indistinguishable set, entanglement can be used as resource. An important objective of the present study is to learn about the efficient ways of resource sharing among the parties. We also find an interesting application of multipartite product states, which are LOCC indistinguishable in a particular $k$ partition. Starting from such product states, we constitute a protocol to distribute bound entanglement between two spatially separated parties by sending a separable qubit.

DOI
10.1103/physreva.101.012311

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Suppose a set of mmm-partite, m\ensuremath≥3m\ensuremath{\ge}3m\ensuremath≥3, pure orthogonal fully separable states is given. We consider the task of distinguishing these states perfectly by local operations and classical communication (LOCC) in different kkk partitions, 1<k<m1<k<m1<k<m. Based on this task, it is possible to classify the sets of product states into different classes. For tripartite systems, a classification of the sets with explicit examples is presented. Few important cases related to the aforesaid task are also studied when the number of parties, m\ensuremath≥4m\ensuremath{\ge}4m\ensuremath≥4. These cases never appear for a tripartite system. However, to distinguish any LOCC indistinguishable set, entanglement can be used as resource. An important objective of the present study is to learn about the efficient ways of resource sharing among the parties. We also find an interesting application of multipartite product states, which are LOCC indistinguishable in a particular kkk partition. Starting from such product states, we constitute a protocol to distribute bound entanglement between two spatially separated parties by sending a separable qubit.
It is known that many sets of orthogonal quantum states, which cannot be distinguished perfectly by local operations and classical communication (LOCC), have been constructed. However, these discussions have left the following open question: How many entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In this paper, we present a local discrimination protocol which uses entanglement more efficiently than teleportation to distinguish one class of orthogonal product states in 3⊗3⊗3. Then, we generalize it for 3⊗3⊗...⊗3 quantum systems which has 3n parties. For the open question, our results offer rather general insight into why entanglement is useful for such tasks, and present a better understanding of the relationship between entanglement and nonlocality.
A set of orthogonal quantum states is said to be locally indistinguishable if they cannot be perfectly distinguished by local operations and classical communication (LOCC). Otherwise, the states are locally distinguishable. Interestingly, locally indistinguishable states can have productive applications in quantum information processing protocols. In this sense, locally indistinguishable states are useful. On the other hand, it is usual to consider that locally distinguishable states are useless. Nevertheless, recent works suggest that locally distinguishable states should be given due consideration as in certain situations these states can be converted to locally indistinguishable states under orthogonality-preserving LOCC (OP-LOCC). Such a counterintuitive phenomenon motivates us to ask when the aforesaid conversion is possible and when it is not. In this work, we provide different structures of locally distinguishable product and entangled states which do not allow the aforesaid conversion. We also provide certain structures of locally distinguishable states which allow the aforesaid conversion. In this way, we classify the locally distinguishable sets by introducing hierarchies among them. In a multipartite system, this study becomes more involved as there exist multipartite locally distinguishable sets which cannot be converted to locally indistinguishable sets by OP-LOCC across any bipartition. We say this as ``no activation across bi-partitions".
Locally indistinguishable states are useful to distribute information among spatially separated parties such that the information is locked. This implies that the parties are not able to extract the information completely via local operations and classical communication (LOCC), while it might be possible via LOCC when the parties share entanglement. In this work, we consider an information distribution protocol using orthogonal states for $m\ensuremath{\ge}3$ spatially separated parties such that even if any $k\ensuremath{\le}(m\ensuremath{-}1)$ parties collaborate still the information cannot be revealed completely. Such a protocol is useful to understand up to what extent the encoded information remains locked. However, if required, the parties can share entanglement and extract the information completely by LOCC. To make the process resource efficient, it should consume less number of entangled states. We show that though the set of states, which are locally indistinguishable across every bipartition, are sufficient for the above protocol, they may consume a higher number of entangled states when aiming for complete information extraction. We establish this by constructing a class of locally indistinguishable sets of orthogonal states which can be employed to accomplish the above protocol and these sets consume less number of entangled states, compared with the former sets, for complete information extraction. In fact, this difference in the number of required entangled states for complete information extraction grows linearly with the number of parties. This study sheds light on suitable use of local indistinguishability property of quantum states as a resource, and thus we demonstrate an efficient way of information distribution.
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.
It is known that there are many sets of orthogonal product states which cannot be distinguished perfectly by local operations and classical communication (LOCC). However, these discussions have left the following open question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In m ⊗ n, certain classes of unextendible product bases (UPB) which can be distinguished perfectly using entanglement as a resource, had been presented in 2008. In this paper, we present protocols which use entanglement more efficiently than teleportation to distinguish some classes of orthogonal product states in m ⊗ n, which are not UPB. For the open question, our results offer rather general insight into why entanglement is useful for such tasks, and present a better understanding of the relationship between entanglement and nonlocality.
It is necessary to investigate the local distinguishability of orthogonal quantum state sets, as their adoption in protocol design helps diminish quantum state transmission and cut operational costs. In this paper, we explore the local distinguishability of six orthogonal product states (OPSs) on any bipartite quantum system. We classify different sets of six bipartite OPSs into eight categories by using the vectors of the numbers of pairwise orthogonality relations, where any two states are orthogonal on only one subsystem within each set. We find that these eight categories contain a total of 78 distinct cases, all but five of which are perfectly distinguishable via local operations and classical communication (LOCC). Furthermore, we discuss the local distinguishability of those five distinct cases in detail. Our work explicitly characterizes the local distinguishability of six bipartite OPSs.