Locally distinguishing unextendible product bases by using entanglement efficiently

Abstract

Any set of states which cannot be perfectly distinguished by local operations and classical communication (LOCC) alone, can always be locally distinguished using quantum teleportation with enough entanglement resource. However, in quantum information theory, entanglement is a very valuable resource, so it leaves the following open question: how to accomplish this task more efficiently than teleportation, that is, design the local discrimination protocol using less entanglement resource. In this paper, we first present two protocols to locally distinguish a set of unextendible product bases (UPB) in 5 ⊗ 5 by using different entanglement resource. Then, we generalize the distinguishing methods for a class of UPB in d ⊗ d, where d is odd and d 3. Furthermore, for a class of UPB in d ⊗ d, where d is even and d 4, we prove that these states can also be distinguished by LOCC with multiple copies of low-dimensional entanglement resource. These results offer rather general insight into how to use entanglement resource more efficiently, and also reveal the phenomenon of less nonlocality with more entanglement.

Strongly nonlocal unextendible product bases do exist

“…It costs 2 log 2 d ebits of entanglement resource in the above discrimination protocol. Protocols consuming less entanglement than the teleportationbased protocol attract much attention in recent years [37,43,47,51,53].…”

Strong quantum nonlocality from hypercubes

“…One can use entanglement as a resource to finish the locally distinguished protocol when the given set is locally indistinguishable. Such protocol is called the entanglement-assisted discrimination [29,[32][33][34][35][36][37][38][39][40]. It is also to quantify how many entanglement resource are needed to local distinguish the strongly nonlocal set constructed here in all bipartitions.…”

Unextendible product bases from tile structures and their local entanglement-assisted distinguishability

“…In this section, we provide a method of locally distinguishing UPBs constructed in Proposition 4 assisted by entanglement, because UPBs cannot be distinguished perfectly by LOCC alone [11]. When m = n ≥ 3 are odd, the UPB constructed from Proposition 4 can be perfectly distinguished by LOCC with a ⌈ m 2 ⌉ ⊗ ⌈ m 2 ⌉ maximally entangled state [17]. We will prove that any UPB from Proposition 4 can be perfectly distinguished by LOCC with a ⌈ m 2 ⌉ ⊗ ⌈ m 2 ⌉ maximally entangled state in Theorem 7.…”

“…Then, local distinguishability with entanglement as a resource attracted more and more attention [12][13][14][15][16]. Recently, it has been shown that some UPBs constructed from tile structures in C m ⊗ C m can be distinguished by LOCC with a ⌈ m 2 ⌉⊗⌈ m 2 ⌉ maximally entangled state when m ≥ 3 is odd [17,18]. In particular, Ref.…”