Entanglement as a resource to distinguish orthogonal product states

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 fact, when the parties share enough entanglement, any set of locally indistinguishable states can always be distinguished by LOCC. That is, with enough entanglement, LOCC can be used to teleport 40 the full multipartite state to a single party, and this party can then make a measurement to determine which state they were given. For example, in a bipartite quantum system m ⊗ n(2 ≤ m ≤ n), a maximally entangled state Ψ = ∑ = − i i m i m 1 0 1 is sufficient for distinguishing any set of orthogonal bipartite quantum states. This can be easily understood by imagining two spatially separated observers Alice and Bob: First, Alice uses the extra shared entanglement to teleport her qudit to Bob; then, this allows Bob, who now holds both qudits, to perform the desired measurement.
However, in recent years, entanglement has been shown to be a valuable resource 41,42 , allowing remote parties to communicate in ways which were previously not thought possible. Examples include the well-known protocols of teleportation 40 , dense coding 43 and data hiding 44,45 . These results have caused a rapid growth in the new field of quantum information. In addition, people believe that discovering the potential power of quantum computers 46 may also rely on entanglement. Thus, entanglement is a valuable resource and it also become important to ask if this task can be accomplished more efficiently.
In 2008, Cohen presented that certain classes of unextendible product bases (UPB) in m ⊗ n(m ≤ n) can be distinguished perfectly with a ⊗ ⌈ ⌉ ⌈ ⌉ m m /2 /2 maximally entangled state 47 . A natural question to ask is whether this task can be completed more efficiently for the general locally indistinguishable orthogonal product states in m ⊗ n, which are not UPB.
In this paper, we will consider the locally indistinguishable orthogonal product states in the general bipartite quantum systems and present the LOCC protocols which, using entanglement as a resource, distinguish these states considerably more efficiently than teleportation. Specifically, in 5 ⊗ 5, we show a set of locally indistinguishable orthogonal product states can be distinguished by LOCC with a 2 ⊗ 2 maximally entangled state. Furthermore, for several classes of locally indistinguishable orthogonal product states on a higher-dimensional m ⊗ n, we present that only needing a 2 ⊗ 2 maximally entangled state, these states are also distinguishable by LOCC. Each of the locally indistinguishable orthogonal product states that we consider can be extended to a complete "nonlocality without entanglement" basis, and the latter can also be distinguished by the same (or slightly altered) protocols. Our results show that the locally indistinguishable quantum states may become distinguishable with a small amount of entanglement resources. And these results also present a better understanding of the relationship between entanglement and nonlocality.

Results
Local distinguishability of orthogonal product states in 5 ⊗ 5. In this section, we present that a set of locally indistinguishable orthogonal product states in 5 ⊗ 5, can be perfectly distinguished by LOCC with a 2 ⊗ 2 maximally entangled state. In 5 ⊗ 5, the following 21 orthogonal product states cannot be distinguished by LOCC 48 , and the structure of these states is different from ref. 48 only lying in exchange of Alice and Bob. For example, in ref. 48 e f e f e f  In the following, applying the proof method which was presented by Cohen 47 , we present a protocol which uses entanglement more efficiently than teleportation to distinguish these quantum states (1). To be precise, the result of bringing in the ancillary systems in state |Ψ〉 ab , and then operating with B 1 on systems bB, is that each of the initial states is transformed as: Let us now describe how the parties can proceed from here to distinguish the states. Alice makes a seven-outcome projective measurement, and we begin by considering the first outcome, The only remaining possibility is φ ′ 9 , which has thus been successfully identified. In the same way, Alice can identify φ ′ 10,11,12 by three projectors Using a rank-1 projector A 5 = |0〉 a 〈0| ⊗ |4〉 A 〈4| onto the Alice's Hilbert space, it leaves φ ′ 5,6,7,8 and annihilates other states in (3). Then, Bob can easily distinguish these four remaining states by projectors onto |0 ± 1〉 B and |2 ± 3〉 B .
Using a rank-2 projector  4 and annihilates other states. Then, Alice can easily distinguish these two remaining states by projectors onto |2 ± 3〉 A . In the same way, we can easily distinguish φ ′ 19,21 .
Scientific RepoRts | 6:30493 | DOI: 10.1038/srep30493 Alice's last outcome is a rank-3 projector onto the remaining part of Alice's Hilbert space , it leaves the last two states φ ′ 13,14 which are orthogonal. In 23 , we know that any two orthogonal states can be distinguished by LOCC. Thus, the two states are locally distinguishable.
For operating with B 2 on systems bB, it creates new states which differ from the states (3) only by ancillary systems |00〉 ab → |11〉 ab and |11〉 ab → |00〉 ab . Thus, the latter can be handled using the exact same method as described above for B 1 .
That is, we have succeeded in designing a protocol that perfectly distinguishes the states (1) using LOCC with an additional resource of a two-qubit maximally entangled state. This completes the proof.
In addition, a 2 ⊗ 2 maximally entangled state is necessary to distinguish the set of product states for the above protocol. When a partially entangled state, |Ψ〉 ab = m|00〉 + n|11〉(m ≠ n), is shared, the states |φ 13,14 〉 will be not orthogonal after Bob performs a two-outcome measurement. This means that 1 ebit is necessary for the protocol. Then, it is interesting to design a protocol to distinguish these states with less than one ebit of entanglement, or to prove that there is no any protocol to accomplish it.
Local distinguishability of orthogonal product states in m ⊗ n. In this section, we consider the set of locally indistinguishable orthogonal product states in m ⊗ n 48 , and the structure of these states is also different from ref. 48 only lying in exchange of Alice and Bob. In the following, we separate it into three cases (2k + 1) ⊗ (2l), (2k) ⊗ (2l) and (2k + 1) ⊗ (2l + 1).
Orthogonal product states in (2k + 1) ⊗ (2l). In this subsection, we first present the following locally indistinguishable orthogonal product states in (2k + 1) ⊗ (2l) 48 . Then, we show that these states can be perfectly distinguished by LOCC with a 2 ⊗ 2 maximally entangled state.  As Theorem 1, we only need to discuss the projector B 1 . To be precise, the result of bringing in the ancillary systems in state |Ψ〉 ab , and then operating with B 1 on systems bB, is that each of the initial states is transformed as: Then, Alice makes a (2k + 3)-outcome projective measurement. Similarly to the proof of Theorem 1, Alice can Using a rank-1 projector A 2k+1 = |0〉 a 〈0| ⊗ |2k〉 A 〈2k| onto the Alice's Hilbert space, it leaves φ ′ + … + − k k l 2 1, ,2 2 2 and annihilates other states in (6). Then, Bob can easily distinguish these remaining states by projectors onto |i ± (i + 1)〉 B , i = 0, 2, …, 2l − 4.

Orthogonal product states in (2k) ⊗ (2l).
In this subsection, we consider the following locally indistinguishable orthogonal product states in (2k) ⊗ (2l) 48 .   In the following, we present that the above states are LOCC distinguishable with a 2 ⊗ 2 maximally entangled state.
When Alice uses a rank-(2k Similarly to the proof of Theorem 2, these states can also be distinguished by LOCC.
For the last rank-3 projector it leaves the last four states φ ′ + − … + − k l k l 6 6 12, , 6 6 9 . Similarly to the proof of Theorem 2, these states can also be distinguished by LOCC.
Thus, the states (7) can be perfectly distinguished using LOCC with an additional resource of a 2 ⊗ 2 maximally entangled state. This completes the proof.
In (2k + 1) ⊗ (2l + 1), the states (10) are a generalization of the states (1). Thus, the proof is similar to the proof of Theorem 1. In the following, we only give a simple proof.
That is, we have succeeded in designing a protocol to distinguish the states (10) by LOCC with a two-qubit maximally entangled state. This completes the proof.
In the proof, the important idea is that entanglement provides multiple Hilbert space, and that the parties can, independently of one another, act on these subspaces in ways that differ from one subspace to the next. This allows an apart of Hilbert space such that initially nonorthogonal pairs of local states end up being orthogonal, aiding the process of distinguishing the set of states. It should be noted that our protocols do not rely on details of the individual states, but only on the general way they are distributed through the space.
As Theorem 1, a 2 ⊗ 2 maximally entangled state is also necessary to distinguish these product states in m ⊗ n for the above protocol. Furthermore, it will be good to do the analysis using Ψ = + ⩾ m n m n 00 11 ( ) ab as a resource. In particular, we are interesting in the optimal probability of distinguishing the states using such a state, and whether it is better than 2n 2 , which is the optimal probability of converting such a state into a maximally entangled state, where m⩾n. However, we have not succeeded in doing so. Hence, it remains an open question whether it is possible to distinguish these states with less than one ebit of entanglement.
In addition, each of the locally indistinguishable orthogonal product states that we consider can be extended to a complete "nonlocality without entanglement" basis 48 . Therefore, the latter can also be distinguished by the same (or slightly altered) protocols.

Conclusion
In this paper, we present that the locally indistinguishable orthogonal product states in m ⊗ n, which are constructed by Wang et al. 48 , can be perfectly distinguished by LOCC with a 2 ⊗ 2 maximally entangled state. Our results show that the locally indistinguishable quantum states may become distinguishable with a small amount of entanglement resources. And we hope that these results can lead to a better understanding for the relationship between nonlocality and entanglement. Recently, Bandyopadhyay et al. 49 explore the question of entanglement as a universal resource for implementing quantum measurements by LOCC. And they show that for most multipartite systems (consisting of three or more subsystems), there is no entangled state from the same space that can enable all measurements by LOCC. Thus, it is also interesting to look for entanglement as a resource to locally distinguish multipartite quantum states.