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arXiv:quant-ph/0304027 · Physical Review A

Distinguishability of complete and unextendible product bases

S. De Rinaldis

Published Aug 1, 2004• 102 citations
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Abstract

It is not always possible to distinguish multipartite orthogonal states if only local operations and classical communication (LOCC) are allowed. We prove that we cannot distinguish the states of an unextendible product basis by LOCC even with infinite resources (infinite-dimensional ancillas, infinite number of operations). Moreover we give a necessary and sufficient condition for the LOCC distinguishability of a complete product basis.

DOI
10.1103/physreva.70.022309

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arXiv:quant-ph/0510198•Oct 26, 2005•2 citations
It is not always possible to distinguish multipartite orthogonal states if only local operations and classical communication (LOCC) are allowed. We prove that we cannot distinguish the states of an unextendible product basis by LOCC even with infinite resources (infinite-dimensional ancillas, infinite number of operations). Moreover we give a necessary and sufficient condition for the LOCC distinguishability of a complete product basis.
It is known that the states in an unextendible product basis (UPB) cannot be distinguished perfectly when the parties are restricted to local operations and classical communication (LOCC). Previous discussions of such bases have left open the following question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In this paper, I present protocols which use entanglement more efficiently than teleportation to distinguish certain classes of UPB's. The ideas underlying my approach to this problem offer rather general insight into why entanglement is useful for such tasks.
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.
In a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on CdA×dA⊗CdB×dB\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}CdA​×dA​⊗CdB​×dB​, we strengthen this existence result by constructing an O(dA2dB2)O(d_A^2 d_B^2)O(dA2​dB2​)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.
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.
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.
Now, the known ensembles of orthogonal states which are distinguishable by local operators and classical communication (LOCC) satisfy the condition that the sum of Schmidit numbers of the orthogonal states is not bigger than the dimensions of the whole space. A natural question is whether an arbitary ensembles of LOCC-distinguishable orthogonal states satisfies the condition. We first show that, in this paper, the answer is positive. Then we generalize it into multipartite systems, and show that a necessary condition for LOCC-distinguishability of multipartite orthogonal quantum states is that the sum of the least numbers of the product states (For bipartite system, the least number of product states is Schmidit number) of the orthogonal states is not bigger than the dimensions of the Hilbert space of the multipartite system. This necessary condition is very simple and general, and one can get many cases of indistinguishability by it. It means that the least number of the product states acts an important role in distinguishablity of states, and implies that the least number of the product states may be an good manifestion of quantum nonlocality in some sense. In fact, entanglement emphases the "amount" of nonlocality, but the least number of the product states emphases the types of nonlocality. For example, the known W states and GHZ states have different least number of the product states, and are different in type.