Local distinguishability of orthogonal quantum states with multiple

PHYSICAL REVIEW A 97, 022334 (2018)

Local distinguishability of orthogonal quantum states with multiple copies of 2 ⊗ 2 maximally entangled states Zhi-Chao Zhang,1,2 Yan-Qi Song,1 Ting-Ting Song,3 Fei Gao,1,* Su-Juan Qin,1 and Qiao-Yan Wen1 1

State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China 2 State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China 3 Department of Computer Science, College of Information Science and Technology, Jinan University, Guangzhou 510632, China (Received 27 December 2017; published 23 February 2018) 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 ⊗ 5 by using two copies of 2 ⊗ 2 maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in d ⊗ d (d is odd). Furthermore, for a class of nonlocality of orthogonal product states in d ⊗ d (d 5), we prove that these states can be distinguished by LOCC with two copies of 2 ⊗ 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 ⊗ 2 maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement. DOI: 10.1103/PhysRevA.97.022334 I. INTRODUCTION

In recent years, quantum information theory has received wide attention [1–7]. One of the main goals is to study the problem of local distinguishability of quantum states [8–19]. When a set of orthogonal quantum states cannot be distinguished by local operations and classical communication (LOCC), it reflects the fundamental feature of quantum mechanics called nonlocality [20–28]. For instance, Bennett et al. discovered that there exist 3 ⊗ 3 pure orthogonal product bases that are indistinguishable by LOCC and showed the phenomenon of nonlocality without entanglement, which is a fundamental result in distinguishing quantum states [1]. Various related results have been presented [23–28]. It is well known that the discrimination of orthogonal quantum states is difficult with only LOCC. However, the presence of additional entanglement can help to overcome such a restriction [29]. Using enough entanglement, we can teleport [30] the full multipartite state to a single party by LOCC; then this party can determine which state they were given. For example, in m ⊗ n (2 m n), a maximally entangled state | = √1m m−1 i=0 |i|i is sufficient to distinguish any sets of orthogonal bipartite quantum states. This is because Alice can first use the extra shared entanglement to teleport her qudit to Bob; then Bob has the set of orthogonal bipartite quantum states to perform the desired measurement. However, entanglement is a very valuable resource [31,32], allowing remote parties to communicate in ways which were previously not thought possible, for example, the well-known protocols of teleportation [30], dense coding [33], and data hiding [34,35]. In addition, entanglement may be used to discover

*

[email protected]

2469-9926/2018/97(2)/022334(7)

the potential power of quantum computers [36]. Therefore, entanglement is a valuable resource and an interesting question that remains to be answered is whether this task can be accomplished more efficiently [37–41]. Cohen showed that certain classes of unextendible product bases (UPBs) in m ⊗ n (m n) can be locally distinguished by using a m/2 ⊗ m/2 maximally entangled state [29]. Recently, Bandyopadhyay et al. proved that for most systems consisting of three or more subsystems, there is no entangled state from the same space that can enable all measurements by LOCC [39]. A natural question to ask is whether a similar task can be completed more efficiently with two or more copies of a low-dimensional entanglement resource. The main contribution of this paper is to answer the above question in the positive. Specifically, for a set of locally indistinguishable orthogonal product states in 5 ⊗ 5, we present a distinguishing protocol by using two copies of 2 ⊗ 2 maximally entangled states. Then we generalize the distinguishing method for a class of locally indistinguishable orthogonal product states in d ⊗ d (d is odd) with multiple copies of 2 ⊗ 2 maximally entangled states. Different from the previous high-dimensional entanglement resource, we only use multiple copies of the 2 ⊗ 2 entanglement resource to complete this task. Thus, our method should be easier to implement in a real experiment because it only needs one piece of equipment which can produce 2 ⊗ 2 maximally entangled states instead of highdimensional entangled states which will change for different sets of quantum states. Furthermore, for a class of nonlocality of orthogonal product states in d ⊗ d (d 5), we prove that these states are locally distinguishable with two copies of 2 ⊗ 2 maximally entangled states. Finally, we construct two classes of LOCC indistinguishable multipartite orthogonal product states based on the above bipartite orthogonal product states and prove that these states can also be distinguished by LOCC with multiple copies of 2 ⊗ 2 maximally entangled states.

022334-1

©2018 American Physical Society

ZHANG, SONG, SONG, GAO, QIN, AND WEN

0 0 1

Bob 2

1 1,2


3

4 5,6

B1 = (|00bB 00| + |01bB 01| + |02bB 02| 19,20

+ |03bB 03| + |14bB 14|) ⊗ Id ,

25

B2 = (|10bB 10| + |11bB 11| + |12bB 12|

7,8

23,24

+ |13bB 13| + |04bB 04|) ⊗ Id .

21,22 13,14

4

11,12

(2)

For operating with B1 on systems bBd, each of the initial states is transformed into

9,10

|φi = |φi |00ab |cd , i = 1, . . . ,4,11, . . . ,25

FIG. 1. Structure of 25 orthogonal product basis quantum states in 5 ⊗ 5.

|φi = |φi |11ab |cd , i = 5,6,7,8, |φ9,10

Our results show that the locally indistinguishable quantum states may become distinguishable with a small amount of entanglement resource. These results can also let us have a better understanding of the relationship between entanglement and nonlocality. In the following, using multiple copies of 2 ⊗ 2 maximally entangled states, we present in Sec. II a method to locally distinguish two classes of bipartite orthogonal product states. Then, in Sec. III, we generalize it to multipartite quantum systems. Finally, we end with a summary and discussion. II. LOCAL DISTINGUISHABILITY OF BIPARTITE ORTHOGONAL QUANTUM STATES

In this section we present a method to locally distinguish two classes of orthogonal product states in d ⊗ d with multiple copies of 2 ⊗ 2 maximally entangled states. First, we show that a set of orthogonal product states in 5 ⊗ 5 can be locally distinguished by using two copies of 2 ⊗ 2 maximally entangled states. From Ref. [24] we know the following 25 LOCC indistinguishable orthogonal product states, which have the structure of Fig. 1: 1 |i ± j = √ (|i ± |j ), 0 i < j 4 2 |φ1,2 = |0A |0 ± 1B , |φ3,4 = |0A |2 ± 3B , |φ5,6 = |0 ± 1A |4B , |φ7,8 = |2 ± 3A |4B , |φ9,10 = |4A |3 ± 4B , |φ11,12 = |4A |1 ± 2B , |φ13,14 = |3 ± 4A |0B , |φ15,16 = |1 ± 2A |0B , |φ17,18 = |1A |1 ± 2B , |φ19,20 = |1 ± 2A |3B , |φ21,22 = |3A |2 ± 3B , |φ23,24 = |2 ± 3A |1B , |φ25 = |2A |2B .

+

+ |11). Then Bob performs a two|11) and |cd = outcome measurement, each outcome corresponding to a rank5 projector:

17,18 15,16

√1 (|00 2

√1 (|00 2

3,4

Alice 2 3

Proof. First of all, Alice and Bob share |ab =

(1)

Now, with the help of two copies of 2 ⊗ 2 maximally entangled states, we study the local distinguishability of these quantum states (1). Theorem 1. In 5 ⊗ 5, the above 25 states can be perfectly distinguished by LOCC with two copies of 2 ⊗ 2 maximally entangled states.

(3)

= (|43AB |00ab ± |44AB |11ab )|cd .

For operating with B2 on systems bBd, it creates new states which differ from the states (3) only by ancillary systems |00ab → |11ab and |11ab → |00ab . Then the latter can be handled using the exact same method as B1 . Thus, we only need to discuss B1 . Let us now describe how the parties can proceed from here to distinguish these states. Alice makes a six-outcome projective measurement, and we begin by considering the first outcome A1 = |1a 1| ⊗ |0 + 1A 0 + 1| ⊗ Ic . The only remaining possibility is |φ5 , which has thus been successfully identified. In the same way, Alice can identify by three projectors A2 = |1a 1| ⊗ |0 − 1A 0 − 1| ⊗ |φ6,7,8 Ic , A3 = |1a 1| ⊗ |2 + 3A 2 + 3| ⊗ Ic , and A4 = |1a 1| ⊗ |2 − 3A 2 − 3| ⊗ Ic , respectively. Using a rank-1 projector A5 = |0a 0| ⊗ |0A 0| ⊗ Ic on Alice’s Hilbert space leaves |φ1,2,3,4 and annihilates other states in (3). Then Bob can easily discriminate the four remaining states by projecting onto |0 ± 1B and |2 ± 3B . Alice’s last outcome is a rank-5 projector on the remaining part of Alice’s Hilbert space A6 = (|0a 0| ⊗ |1A 1| + |0a 0| ⊗ |2A 2| + |0a 0| ⊗ |3A 3| + |0a 0| ⊗ |4A 4| + |1a 1| ⊗ |4A 4|) ⊗ Ic . This leaves |φ9,10,...,25 and annihilates other states. Then, when Bob uses the projector B61 = |0b 0| ⊗ |0B 0| ⊗ Id , it , which can be easily distinguished by leaves |φ13,14,15,16 projectors on |1 ± 2A and |3 ± 4A . When Bob uses a projector B62 = (|0b 0| ⊗ |1B 1| + |0b 0| ⊗ |2B 2| + |0b 0| ⊗ |3B 3| + |1b 1| ⊗ |4B 4|) ⊗ Id , it can leave |φ9,10,11,12,17,...,25 and annihilate other states. Then Alice uses A621 = (|0a 0| ⊗ |4A 4| + |1a 1| ⊗ |4A 4|) ⊗ Ic , which leaves |φ9,10,11,12 . Bob can easily distinguish |φ11,12 by projectors on |1 ± 2B and leave the last two states |φ9,10 , which are orthogonal. From Ref. [2] we know that any two orthogonal states can be distinguished by LOCC. Thus, the two states are locally distinguishable. For A622 = (|0a 0| ⊗ |1A 1| + |0a 0| ⊗ |2A 2| + |0a 0| ⊗ |3A 3|) ⊗ Ic , it leaves |φ17,...,25 , which can be locally distinguished with |cd = √12 (|00 + |11) [29]. That is to say, we have succeeded in designing a protocol to perfectly distinguish the states (1) using LOCC with two copies of the 2 ⊗ 2 maximally entangled state. This completes the proof.

022334-2

LOCAL DISTINGUISHABILITY OF ORTHOGONAL …

0

Bob

1

0


B2 = [|10b1 B 10| + |11b1 B 11| + · · ·

n-2

n-1

n 2, n 1

1, 2

+ |1(d − 2)b1 B 1(d − 2)| + |0(d − 1)b1 B 0(d − 1)|] ⊗ Ib2 ⊗ · · · ⊗ Ib(d−1)/2 .

n, n 1

1 2

4n 5,

|φi = |φi |00a1 b1 |a2 b2 · · · |a(d−1)/2 b(d−1)/2 , n2

n

2

i = 1, . . . ,d − 1,2d + 1, . . . ,d 2

7, n 2 8

n 2 1,

n2

n

2

5,

n

2

6

n-1

|φi = |φi |11a1 b1 |a2 b2 · · · |a(d−1)/2 b(d−1)/2 , i = d,d + 1, . . . ,2d − 2

2

n 2 3, n 2

n-2

Like Theorem 1, we only need to discuss B1 , which transforms the initial states into

4n 4

Alice

(5)

3n 2, 3n 1

|φ2d−1,2d

4

2n 3, 2n 2

= [|(d − 1)(d − 2)AB |00a1 b1 ± |(d − 1)(d − 1)AB |11a1 b1 ] × |a2 b2 · · · |a(d−1)/2 b(d−1)/2 .

(6)

Then Alice makes the next projective measurement: 3n 4,3n 3

2n 1, 2n

A1 = |1a1 1| ⊗ |0 + 1A 0 + 1| ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 , A2 = |1a1 1| ⊗ |0 − 1A 0 − 1| ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 ,

FIG. 2. Structure of n2 orthogonal product basis quantum states in n ⊗ n (n is odd).

In the following, we consider a more general quantum system d ⊗ d (d is odd). Reference [24] also found that these states are LOCC indistinguishable and have the structure of Fig. 2: 1 |i ± j = √ (|i ± |j ), 0 i < j n − 1 2

Ad−1 = |1a1 1| ⊗ |(d − 3) − (d − 2)A (d − 3) − (d − 2)| ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 , Ad = |0a1 0| ⊗ |0A 0| ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 , Ad+1 = [|0a1 0| ⊗ |1A 1| + |0a1 0| ⊗ |2A 2| + · · · ⊗ |(d − 1)A (d − 1)|] ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 .

.. . (4)

Similar to Theorem 1, we have the next result. Theorem 2. In d ⊗ d (d is odd), the above structure of d 2 states is locally distinguishable by using d−1 copies of 2 ⊗ 2 2 maximally entangled states. Proof. For d = 3, Cohen exhibited that nine states can be locally distinguished with 2 ⊗ 2 maximally entangled states [29]. This was presented in Theorem 1 for d = 5. Now we consider d = n, where n is odd. First suppose that when n = d − 2, the above structure of (d − 2)2 states can be perfectly distinguished by LOCC with d−3 copies of 2 ⊗ 2 maximally 2 entangled states. Next we show that when n = d, the above structure of d 2 states can be locally distinguished by using d−1 2 copies of 2 ⊗ 2 maximally entangled states. Similarly, let Alice and Bob share d−1 2 ⊗ 2 maximally 2 . entangled states |ai bi = √12 (|00 + |11), i = 1,2, . . . , d−1 2 Then Bob performs the following two-outcome measurement: B1 = [|00b1 B 00| + |01b1 B 01| + · · · + |0(d − 2)b1 B 0(d − 2)| + |1(d − 1)b1 B 1(d − 1)|] ⊗ Ib2 ⊗ · · · ⊗ Ib(d−1)/2 ,

.. .

+ |0a1 0| ⊗ |(d − 1)A (d − 1)| + |1a1 1|

|φ1,2 = |0A |0 ± 1B , |φ3,4 = |0A |2 ± 3B ,

|φn2 = |(n − 1)/2A |(n − 1)/2B .

A3 = |1a1 1| ⊗ |2 + 3A 2 + 3| ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 ,

(7)

, For Ai , i = 1, . . . ,d − 1, Alice can identify |φd,d+1,...,2d−2 respectively. For Ad , it leaves |φ1,2,...,d−1 , which can be easily distinguished by Bob. For Ad+1 , it only needs to distinguish |φ2d−1,2d,...,d Next Bob uses 2 . B(d+1)1 = |0b1 0| ⊗ |0B 0| ⊗ Ib2 ⊗ · · · ⊗ Ib(d−1)/2 to get |φ3d−2,...,4d−4 , which can be distinguished by Alice. For B(d+1)2 = [|0b1 0| ⊗ |1B 1| + |0b1 0| ⊗ |2B 2| + · · · + |0b1 0| ⊗ |(d − 2)B (d − 2)| + |1b1 1| ⊗ |(d − 1)B (d − states 1)|] ⊗ Ib2 ⊗ · · · ⊗ Ib(d−1)/2 , |φ2d−1,...,3d−3,4d−3,...,d 2 need to be distinguished. Then Alice performs A(d+1)21 = [|0a1 0| ⊗ |(d − 1)A (d − 1)| + |1a1 1| ⊗ |(d − 1)A (d − 1)|] ⊗ Ia2 ⊗ · · · ⊗ Ia(d−1)/2 and gets |φ2d−1,...,3d−3 . For these states, Bob can easily discriminate |φ2d+1,...,3d−3 by projectors on |1 ± 2B , . . . ,|(d − 4) ± (d − 3)B , and orthogonal can be locally distinguished [2]. Finally, states |φ2d−1,2d there are only |φ4d−3,...,d 2 states that need to be further distinguished. Based on previous evidence, these states are locally distinguishable. Therefore, with d−1 copies of 2 ⊗ 2 2 maximally entangled states, the above structure of d 2 states can be locally distinguished. This completes the proof. Next we consider a class of different orthogonal product states in d ⊗ d (d 5) and prove that our method is also valid for these states. First, we add the state |φ4d−3 = |(d − 1)A |2 + 3B to these states in Ref. [25], and the new set of states are also orthogonal. Thus, we know that the following 4d − 3 states are LOCC indistinguishable in d ⊗ d (d 5)

022334-3



± |(d − 1)1AB |00ab |11cd ,

[25]. Here we add one state in order to construct multipartite orthogonal product states in the next section, 1 |m ± n = √ (|m ± |n), 0 m < n d − 1 2 |φi = |iA |0 − iB ,

|φd−1,2d−2

|φi+2d−2 = |0 − iA |j B , |φi+3d−3 = |0 + iA |j B , |φ4d−3 = |(d − 1)A |2 + 3B ,

(8)

where i = 1, . . . ,d − 2; j = i + 1; and only for i = d − 1, j = 1. Then, using two copies of the 2 ⊗ 2 entanglement resource, we present the local distinguishability of the above 4d − 3 states. Theorem 3. In d ⊗ d (d 5), the above 4d − 3 states are LOCC distinguishable with two copies of 2 ⊗ 2 maximally entangled states. Proof. Similarly, Alice and Bob need to share two 2 ⊗ 2 maximally entangled states |ab = √12 (|00 + |11)

and |cd = √12 (|00 + |11). Then Bob performs the following measurement: B1 = [|00bB 00| + |01bB 01| + · · · + |0(d − 2)bB × 0(d − 2)| + |1(d − 1)bB 1(d − 1)|] ⊗ Id , B2 = [|10bB 10| + |11bB 11| + · · · + |1(d − 2)bB × 1(d − 2)| + |0(d − 1)bB 0(d − 1)|] ⊗ Id .

(9)

In the same way, we only consider B1 , which can create new states as follows: = |φi |00ab |cd ,

i = 1, . . . ,4d − 3, i = d − 1,2d − 2,3d − 4,4d − 5 |φi = |φi |11ab |cd , i = 3d − 4,4d − 5 = [|(d − 1)0AB |00ab |φd−1,2d−2

± |(d − 1)(d − 1)AB |11ab ]|cd .

(10)

Like Theorem 2, we can easily distinguish {|φi ,i = 1, . . . ,4d − 3,i = 1,d,2,d + 1,d − 1,2d − 2,2d − 1,3d − 2,2d,3d − 1,3d − 3,4d − 4,4d − 3} and leave the last 13 states {|φi ,i = 1,d,2,d + 1,d − 1,2d − 2,2d − 1,3d − 2,2d,3d − 1,3d − 3,4d − 4,4d − 3} to be distinguished. Here we need another entanglement resource |cd to help and Alice makes the next measurement:

In this section we first construct a set of orthogonal product states in 3 ⊗ 3 ⊗ 3 ⊗ 3 and prove that these states are LOCC indistinguishable. Then we show that these states can be locally distinguished with two copies of 2 ⊗ 2 maximally entangled states. Furthermore, we generalize this result in 3 ⊗ 3 ⊗ · · · ⊗ 3, which has 2n parties. Then we show that a class of orthogonal product states in d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , for dj odd with j = 1,2, . . . ,n, can be locally distinguished by using d1 +d2 +···+dn −n copies of 2 ⊗ 2 maximally entangled states. 2 Finally, we also construct a class of orthogonal product states in d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , dj 5, j = 1,2, . . . ,n, which can be distinguished by LOCC with 2n copies of 2 ⊗ 2 maximally entangled states. In 3 ⊗ 3, we know the following quantum states, which were shown by Bennett et al. to be indistinguishable by LOCC [1]: 1 |i ± j = √ (|i ± |j ), 0 i < j 2 2 |φ1,2 = |0A |0 ± 1B , |φ3,4 = |0 ± 1A |2B ,

A2 = (|10cA 10| + |11cA 11| + · · · + |1(d − 2)cA

|φ5,6 = |2A |1 ± 2B , |φ7,8 = |1 ± 2A |0B ,

(11)

We also only consider A1 and the remaining 13 states are transformed into = |φi |00ab |00cd ,

|φ9 = |1A |1B .

(13)

Based on these states, we construct the next orthogonal product states in 3 ⊗ 3 ⊗ 3 ⊗ 3, |ϕi = |φi |φ9 , i = 1,2, . . . ,8

i = 1,d,2,d + 1,2d − 1,3d − 2,2d,3d − 1 |φ3d−3,4d−4

(12)

III. LOCAL DISTINGUISHABILITY OF MULTIPARTITE ORTHOGONAL QUANTUM STATES

×0(d − 2)| + |1(d − 1)cA 1(d − 1)|) ⊗ Ia ,

|φi

= [|(d − 1)0AB |00ab

Then Bob can easily discriminate {|φi ,i = d − 1,2d − 2, 4d − 3}. The rest of the states can be locally distinguished by Alice and Bob as in Theorem 1. Thus, we have succeeded in locally distinguishing the states (8) by using two copies of 2 ⊗ 2 maximally entangled states. This completes the proof. We have presented a different distinguishing method which uses two or more low-dimensional entanglement resources instead of a high-dimensional entanglement resource. We think that the previous method is more efficient and saves resources. In addition, for d = 3,4, we cannot add the state |φ4d−3 = |(d − 1)A |2 + 3B to the states in Ref. [25] because these states will not be existing or orthogonal. When not adding the state, these states in Ref. [25] can be locally distinguished with one or two copies of 2 ⊗ 2 maximally entangled states. This result can be easily proved as Theorem 3. In fact, the above method not only can be used in bipartite quantum systems, but also can be used in multipartite quantum systems. In the following, we will present two classes of orthogonal product states in multipartite quantum systems, which can be locally distinguished with multiple copies of 2 ⊗ 2 maximally entangled states.

A1 = (|00cA 00| + |01cA 01| + · · · + |0(d − 2)cA

×1(d − 2)| + |0(d − 1)cA 0(d − 1)|) ⊗ Ia .

= |φ4d−3 |00ab |11cd , ± |(d − 1)(d − 1)AB |11ab ]|11cd .

|φi+d−1 = |iA |0 + iB ,

|φi

|φ4d−3

= |01AB |00ab |00cd

|ϕi+8 = |φ9 |φi , i = 1,2, . . . ,8. 022334-4

(14)



Using this method in [25,42], we can know that the first party cannot start a nontrivial measurement for {|ϕi ,i = 1,2, . . . ,8}. First, let the first party start with the nontrivial † and nondisturbing measurement Mm Mm , ⎤ ⎡ a00 a01 a02 ⎥ ⎢ Mm† Mm = ⎣a10 a11 a12 ⎦. a20 a21 a22 The postmeasurement states {Mm ⊗ I234 |ϕi ,i = 1, . . . ,8} should be mutually orthogonal. Then, considering the states † |ϕ1,5 , we know 0|Mm Mm |20 + 1|1 + 21|11|1 = 0. † Thus, 0|Mm Mm |2 = 0, i.e., a02 = a20 = 0. Similarly, for the states |ϕ3,5 and |ϕ1,7 , we can know a12 = a21 = a01 = a10 = 0. Then considering the states |ϕ3,4 and |ϕ7,8 , we can get † a00 = a11 = a22 . Therefore, all of measurements Mm Mm are proportional to the identity, meaning that the first party cannot start with a nontrivial measurement. In the same way, the second party cannot start a nontrivial measurement for {|ϕi ,i = 1,2, . . . ,8} and other two party cannot start a nontrivial measurement for {|ϕi+8 ,i = 1,2, . . . ,8}. Thus, we get the following result. Theorem 4. In 3 ⊗ 3 ⊗ 3 ⊗ 3, the above 16 states (14) cannot be perfectly distinguished by LOCC. However, as in Theorem 1, two copies of 2 ⊗ 2 maximally entangled states are also sufficient to locally distinguish these states of (14). Theorem 5. In 3 ⊗ 3 ⊗ 3 ⊗ 3, the above 16 states (14) can be locally distinguished with two copies of 2 ⊗ 2 maximally entangled states. For the proof, we give a simple explication. First, the first party and second party share 2 ⊗ 2 maximally entangled states |ab = √12 (|00 + |11) and the other two parties share 2 ⊗ 2 maximally entangled states |cd = √12 (|00 + |11). According to Ref. [29], we know that the first party and second party can locally discriminate {|φi ,i = 1,2, . . . ,9} with |ab = √12 (|00 + |11). Thus, they can identify the states |ϕi according to |φi ,i = 1,2, . . . ,8. When they identify |φ9 , it leaves {|ϕi ,i = 9,10, . . . ,16} to be distinguished. For the rest of the states, the other two parties can also distinguish them with |cd = √12 (|00 + |11). Therefore, these states can be perfectly distinguished by LOCC with two copies of 2 ⊗ 2 maximally entangled states. In the following, we generalize the above results in 3 ⊗ 3 ⊗ · · · ⊗ 3, which has 2n parties. First, we construct the next

orthogonal product states in 3 ⊗ 3 ⊗ · · · ⊗ 3, which has 2n parties, |ϕi = |φi 12 |φ9 34 · · · |φ9 (2n−1)2n , |ϕi+8 = |φ9 12 |φi 34 · · · |φ9 (2n−1)2n , .. . |ϕi+8j = |φ9 12 · · · |φi (2j +1)(2j +2) · · · |φ9 (2n−1)2n , .. . |ϕi+8(n−1) = |φ9 12 · · · |φ9 (2n−3)(2n−2) |φi (2n−1)2n , i = 1,2, . . . ,8, j = 2, . . . ,n − 2.

(15)

Similarly to Theorem 4, every party cannot start a nontrivial measurement, so we get the result as follows. Theorem 6. In 3 ⊗ 3 ⊗ · · · ⊗ 3, which has 2n parties, these quantum states (15) cannot be perfectly distinguished by LOCC. In the same way, we study the local distinguishability of these states with multiple copies of the low-dimensional entanglement resource. Theorem 7. In 3 ⊗ 3 ⊗ · · · ⊗ 3, which has 2n parties, these quantum states (15) can be locally distinguished with n copies of 2 ⊗ 2 maximally entangled states. Like Theorem 5, the 2n parties share n copies of 2 ⊗ 2 maximally entangled states |ai bi = √12 (|00 + |11), i = 1,2, . . . ,n. Then {|φi ,i = 1,2, . . . ,9} can be locally distinguished by the first and second parties with |a1 b1 = √1 (|00 + |11) [29]. Thus, |ϕi can be identified by |φi , i = 2 1,2, . . . ,8. After identifying |φ9 , only {|ϕi ,i = 9,10, . . . ,8n} need to be further distinguished. In the same way, with |ai bi = √12 (|00 + |11), {|ϕ8i+j −8 ,j = 1,2, . . . ,8} can be locally distinguished, for i = 2, . . . ,n. Thus, with n copies of 2 ⊗ 2 maximally entangled states, these states are locally distinguishable. For the states of (4), we can also construct locally indistinguishable orthogonal product states in more general multipartite quantum systems. We can prove that these states can be distinguished by LOCC with multiple copies of 2 ⊗ 2 maximally entangled states. In the following, we first construct these states in d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , for dj odd, j with j = 1,2, . . . ,n. Here we select |φi as the state of (4) in 2 dj ⊗ dj , i = 1, . . . ,dj ,

|ϕi = φi1 12 φd22 34 · · · φdn2 (2n−1)2n , i = 1,2, . . . ,d12 − 1 n 2 1 2

n

|ϕi+d12 −1 = φd 2 12 φi 34 · · · φd 2 (2n−1)2n , i = 1,2, . . . ,d22 − 1 n

1

.. .

j +1

|ϕi+d12 +···+dj2 −j = φd12 12 · · · φi (2j +1)(2j +2) · · · φdn2 (2n−1)2n , i = 1,2, . . . ,dj2+1 − 1, j = 2, . . . ,n − 2 n

1

.. .

1

n−1

n

2 2 |ϕi+d12 +···+dn−1 −n+1 = φd 2 12 · · · φd 2 (2n−3)(2n−2) φi (2n−1)2n , i = 1,2, . . . ,dn − 1. 1

n−1

022334-5

(16)



For these states of (16), all parties cannot start a nontrivial measurement as in Theorem 6. Then the following result can be easily shown. Theorem 8. In d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , for dj odd, with j = 1,2, . . . ,n, the above d12 + d22 + · · · + dn2 − n states are locally indistinguishable. According to Theorem 2, we can also get the next result, which is similar to Theorem 7 and can be easily proved as in Theorem 7. Theorem 9. In d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , for dj odd, with j = 1,2, . . . ,n, the above d12 + d22 + · · · + dn2 − n states n −n can be perfectly distinguished by LOCC with d1 +d2 +···+d 2 ⊗ 2 maximally entangled states. 2 Finally, we consider these states of (8) and construct the following 4(d1 + · · · + dn ) − 4n states in d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , dj 5, j = 1,2, . . . ,n, 2 n

· · · φ4d , i = 1,2, . . . ,4d1 − 4 |ϕi = φi1 12 φ4d 2 −3 34 n −3 (2n−1)2n 1 n

2

|ϕi+4d1 −4 = φ4d1 −3 12 φi 34 · · · φ4dn −3 (2n−1)2n , i = 1,2, . . . ,4d2 − 4 .. .

1 j +1

n

· · · φi (2j +1)(2j +2) · · · φ4d , i = 1,2, . . . ,4dj +1 − 4, j = 2, . . . ,n − 2 |ϕi+4d1 −4+···+4dj −4 = φ4d 1 −3 12 n −3 (2n−1)2n .. .

1 n−1

n

φ |ϕi+4d1 −4+···+4dn−1 −4 = φ4d · · · φ4d i (2n−1)2n , i = 1,2, . . . ,4dn − 4, 1 −3 12 n−1 −3 (2n−3)(2n−2)

(17)

j

where |φi is the state of (8) in dj ⊗ dj , i = 1, . . . ,4dj − 3. When distinguishing these states of (17), every party cannot start a nontrivial measurement [25]. Thus, we can present the next result. Theorem 10. In d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , dj 5, j = 1,2, . . . ,n, the above 4(d1 + · · · + dn ) − 4n states are LOCC indistinguishable. Similarly to Theorems 7 and 9, the local distinguishability of these states will change with two copies of 2 ⊗ 2 maximally entangled states. Theorem 11. In d1 ⊗ d1 ⊗ d2 ⊗ d2 ⊗ · · · ⊗ dn ⊗ dn , dj 5, j = 1,2, . . . ,n, the states of (16) can be perfectly distinguished by LOCC with 2n copies of 2 ⊗ 2 maximally entangled states. In addition, different from previous high-dimensional entangled states which will change for different sets of quantum states, our method only needs one piece of equipment which can produce 2 ⊗ 2 maximally entangled states. Thus, our method should be relatively easier to implement in a real experiment. IV. CONCLUSION

In this paper, with entanglement as a resource to distinguish orthogonal quantum states, we present a method based on multiple copies of low-dimensional entanglement resources instead of a high-dimensional entanglement resource. Our results can lead to a better understanding of the relationship between nonlocality and entanglement. Finally, for studying the local distinguishability of orthogonal product states, UPBs play a very important role. Thus, it is interesting to investigate whether our method can be used for some kind of UPB, for example, the usual Tiles UPB [43], which may need multiple

[1] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59, 1070 (1999). [2] J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local Distinguishability of Multipartite Orthogonal Quantum States, Phys. Rev. Lett. 85, 4972 (2000). [3] K. Feng, Unextendible product bases and 1-factorization of complete graphs, Disc. Appl. Math. 154, 942 (2006).

copies of 3 ⊗ 3 or higher-dimensional entanglement resources. What kind of state sets can always be locally discriminated using multiple copies of 2 ⊗ 2 maximally entangled states without teleportation is also interesting. ACKNOWLEDGMENTS

The authors are grateful for the anonymous referee’s suggestions to improve the quality of this paper. This work was supported by the NSFC (Grants No. 61672110, No. 61572081, and No. 61502200) and the BUPT Excellent Ph.D. Students Foundation (Grant No. CX2017403).

[4] R. Y. Duan, Y. Feng, Y. Xin, and M. S. Ying, Distinguishability of quantum states by separable operations, IEEE Trans. Inf. Theory 55, 1320 (2009). [5] F. Gao, B. Liu, W. Huang, and Q. Y. Wen, Postprocessing of the oblivious key in quantum private query, IEEE J. Sel. Top. Quantum Electron. 21, 98 (2015). [6] C. Y. Wei, T. Y. Wang, and F. Gao, Practical quantum private query with better performance in resisting joint-measurement attack, Phys. Rev. A 93, 042318 (2016).

022334-6



[7] B. Liu, F. Gao, and W. Huang, Sci. China Phys. Mech. Astron. 58, 100301 (2015). [8] P.-X. Chen and C.-Z. Li, Distinguishing the elements of a full product basis set needs only projective measurements and classical communication, Phys. Rev. A 70, 022306 (2004). [9] H. Fan, Distinguishability and Indistinguishability by Local Operations and Classical Communication, Phys. Rev. Lett. 92, 177905 (2004). [10] M. Nathanson, Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases, J. Math. Phys. 46, 062103 (2005). [11] J. Watrous, Bipartite Subspaces Having no Bases Distinguishable by Local Operations and Classical Communication, Phys. Rev. Lett. 95, 080505 (2005). [12] H. Fan, Distinguishing bipartite states by local operations and classical communication, Phys. Rev. A 75, 014305 (2007). [13] R. Y. Duan, Y. Xin, and M. S. Ying, Locally indistinguishable subspaces spanned by three-qubit unextendible product bases, Phys. Rev. A 81, 032329 (2010). [14] N. K. Yu, R. Y. Duan, and M. S. Ying, Any 2 ⊗ n subspace is locally distinguishable, Phys. Rev. A 84, 012304 (2011). [15] N. K. Yu, R. Y. Duan, and M. S. Ying, Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States, Phys. Rev. Lett. 109, 020506 (2012). [16] S. Bandyopadhyay, S. Ghosh, and G. Kar, LOCC distinguishability of unilaterally transformable quantum states, New J. Phys. 13, 123013 (2011). [17] S. Bandyopadhyay, Entanglement, mixedness, and perfect local discrimination of orthogonal quantum states, Phys. Rev. A 85, 042319 (2012). [18] A. Cosentino, Positive-partial-transpose-indistinguishable states via semidefinite programming, Phys. Rev. A 87, 012321 (2013). [19] J. Lebl, A. Shakeel, and N. Wallach, Local distinguishability of generic unentangled orthonormal bases, Phys. Rev. A 93, 012330 (2016). [20] M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local Indistinguishability: More Nonlocality with Less Entanglement, Phys. Rev. Lett. 90, 047902 (2003). [21] S. Bandyopadhyay, More Nonlocality with Less Purity, Phys. Rev. Lett. 106, 210402 (2011). [22] J. Walgate and L. Hardy, Nonlocality, Asymmetry, and Distinguishing Bipartite States, Phys. Rev. Lett. 89, 147901 (2002). [23] A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, A framework for bounding nonlocality of state discrimination, Commun. Math. Phys. 323, 1121 (2013). [24] Z.-C. Zhang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Nonlocality of orthogonal product basis quantum states, Phys. Rev. A 90, 022313 (2014). [25] Z.-C. Zhang, F. Gao, S.-J. Qin, Y.-H. Yang, and Q.-Y. Wen, Nonlocality of orthogonal product states, Phys. Rev. A 92, 012332 (2015).

[26] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, Nonlocality of orthogonal product-basis quantum states, Phys. Rev. A 92, 032313 (2015). [27] J. Niset and N. J. Cerf, Multipartite nonlocality without entanglement in many dimensions, Phys. Rev. A 74, 052103 (2006). [28] G.-B. Xu, Q.-Y. Wen, S.-J. Qin, Y.-H. Yang, and F. Gao, Quantum nonlocality of multipartite orthogonal product states, Phys. Rev. A 93, 032341 (2016). [29] S. M. Cohen, Understanding entanglement as resource: Locally distinguishing unextendible product bases, Phys. Rev. A 77, 012304 (2008). [30] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Phys. Rev. Lett. 70, 1895 (1993). [31] S. Bandyopadhyay, G. Brassard, S. Kimmel, and W. K. Wootters, Entanglement cost of nonlocal measurements, Phys. Rev. A 80, 012313 (2009). [32] S. Bandyopadhyay, Entanglement cost of two-qubit orthogonal measurements, J. Phys. A: Math. Theor. 43, 455303 (2010). [33] C. H. Bennett and S. J. Wiesner, Communication via Oneand Two-Particle Operators on Einstein-Podolsky-Rosen States, Phys. Rev. Lett. 69, 2881 (1992). [34] D. P. DiVincenzo, D. W. Leung, and B. M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48, 580 (2002). [35] W. Matthews, S. Wehner, and A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding, Commun. Math. Phys. 291, 813 (2009). [36] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Sci. Comput. 26, 1484 (1997). [37] S. M. Cohen, Local distinguishability with preservation of entanglement, Phys. Rev. A 75, 052313 (2007). [38] Z.-C. Zhang, F. Gao, T.-Q. Cao, S.-J. Qin, and Q.-Y. Wen, Entanglement as a resource to distinguish orthogonal product states, Sci. Rep. 6, 30493 (2016). [39] S. Bandyopadhyay, S. Halder, and M. Nathanson, Entanglement as a resource for local state discrimination in multipartite systems, Phys. Rev. A 94, 022311 (2016). [40] Ö. Güngör and S. Turgut, Entanglement-assisted state discrimination and entanglement preservation, Phys. Rev. A 94, 032330 (2016). [41] S. Halder, On a class of small nonlocal set of n-party orthogonal product states, arXiv:1603.08438. [42] S. X. Yu and C. H. Oh, Detecting the local indistinguishability of maximally entangled states, arXiv:1502.01274. [43] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases, uncompletable product bases and bound entanglement, Commun. Math. Phys. 238, 379 (2003).

022334-7