Bidirectional Quantum Communication Through the Composite GHZ-GHZ Channel

applied sciences

Article Bidirectional Quantum Communication through the Composite GHZ-GHZ Channel

Shuangshuang Shuai, Na Chen * and Bin Yan College of Electronic and Information Engineering, Shandong University of Science and Technology, Qingdao 266590, China; [email protected] (S.S.); [email protected] (B.Y.) * Correspondence: [email protected]; Tel.: +86-0532-86057522

Received: 5 July 2020; Accepted: 6 August 2020; Published: 8 August 2020

Abstract: This paper solved the problem of transmitting quantum bits (qubits) in a multi-hop and bidirectional way. Considering that the Greenberger–Horne–Zeilinger (GHZ) states are less prone to the decoherence eﬀects caused by the surrounding environment, we proposed a bidirectional quantum communication scheme based on quantum teleportation and the composite GHZ-GHZ states. On a multi-hop quantum path, diﬀerent types of GHZ states are previously shared between the adjacent intermediate nodes. To implement qubit transmission, the sender and intermediate nodes perform quantum measurements in parallel, and then send their measurement results and the types of previously shared GHZ states to the receiver independently. Based on the received information, the receiver performs unitary operations on the local particle, thus retrieving the original qubit. Our scheme can avoid information leakage at the intermediate nodes and can reduce the end-to-end communication delay, in contrast to the hop-by-hop qubit transmission scheme.

Keywords: bidirectional quantum communication; GHZ; quantum teleportation

1. Introduction Quantum teleportation is a process of transmitting unknown quantum states between two distant nodes based on entanglement and some auxiliary classical communication. Since Bennett proposed the concept of quantum teleportation [1], where Einstein–Podolsky–Rosen (EPR) pairs are used as an entanglement channel, quantum teleportation has been paid extensive attention in recent years. Some quantum teleportation schemes have been proposed by using Greenberger–Horne–Zeilinger (GHZ) states, mixed W states, and other entangled states as quantum channels [2–5]. To realize qubit transmission in a quantum communication network, Wang et al. [6] proposed the idea of quantum wireless multi-hop communication based on arbitrary EPR pairs and teleportation, where simultaneous entanglement swapping is utilized to reduce the end-to-end quantum communication delay. To overcome the decoherence effects caused by the surrounding environments and take advantage of partially entangled EPR pairs efficiently, Yu et al. [7] proposed a wireless quantum communication scheme. However, hop-by-hop qubit transmission introduces great communication delay and information leakage at the intermediate nodes. After that, Chen et al. [8] proposed a wide area quantum communication network via partially entangled EPR states, where sequential entanglement swapping is exploited. In this scheme, the security of qubit transmission is improved, while end-to-end quantum communication delay is not reduced to a great extent. Compared to two-particle entanglement, multi-particle entangled states are equipped with better entanglement properties, and are more powerful in revealing the nonlocality of quantum physics [9,10]. Zhan and Zou et al. proposed multi-hop quantum teleportation schemes based on W states and GHZ–Bell channels [11,12]. Considering that duplex quantum communication between two arbitrary nodes is crucial to the future quantum networks, Li et al. [13] first proposed the bidirectional controlled quantum

Appl. Sci. 2020, 10, 5500; doi:10.3390/app10165500 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5500 2 of 11 transmission scheme using a ﬁve-qubit entangled state. Since then, some schemes for bidirectional quantum communication have been proposed [14–21]. To address all the issues mentioned above, such as quantum communication delay, information leakage, and duplex quantum communication, in this paper we investigate multi-hop bidirectional quantum communication based on GHZ states, considering that GHZ states are less prone to the decoherence eﬀects caused by the surrounding environment [22]. The main contribution of this work is as follows. We propose a scheme for multi-hop bidirectional quantum communication using the composite GHZ-GHZ channel, where all the nodes perform quantum measurements in parallel. After that, they send the measurement results and the types of previously shared GHZ-GHZ states to the sender and receiver through the classical channel independently. Based on the received information, the sender and receiver perform appropriate unitary operations to recover the original qubit. Our scheme has a shorter delay for the end-to-end quantum communication and avoids information leakage at the intermediate nodes. The rest of this paper is organized as follows. In Section2, one-hop bidirectional quantum communication based on a GHZ-GHZ entangled channel is discussed. In Section3, we investigate two-hop directional quantum communication. In Section4, a scheme for multi-hop bidirectional quantum communication is proposed. Finally, some discussions and conclusions are given in Section5.

2. One-Hop Bidirectional Quantum Communication In this section, we investigate one-hop bidirectional quantum communication through the composite GHZ-GHZ states. Eight types of GHZ states used in quantum communication are given by E E ψ1 = 1 ( 000 + 111 ), ψ2 = 1 ( 001 + 110 ) E √2 | i | i E √2 | i | i ψ3 = 1 ( 010 + 101 ), ψ4 = 1 ( 011 + 100 ) E √2 | i | i E √2 | i | i (1) ψ5 = 1 ( 000 111 ), ψ6 = 1 ( 001 110 ) E √2 | i − | i E √2 | i − | i ψ7 = 1 ( 010 101 ), ψ8 = 1 ( 011 100 ) √2 | i − | i √2 | i − | i These GHZ states can be transformed into each other through unitary operations, such as E E E E ψ2 = X X I ψ1 , ψ3 = I X I ψ1 E ⊗ ⊗ E E ⊗ ⊗ E ψ4 = X I I ψ1 , ψ5 = Z I I ψ1 E ⊗ ⊗ E E ⊗ ⊗ E (2) ψ6 = ZX X I ψ1 , ψ7 = Z X I ψ1 ⊗ ⊗E E ⊗ ⊗ ψ8 = ZX I I ψ1 ⊗ ⊗ ! ! ! 0 1 1 0 1 0 where X = , Z = are Pauli matrices, and I = is the identity matrix. 1 0 0 1 0 1 − Assume that Alice intends to transmit an arbitrary qubit χ to Bob, while Bob intends to transmit | ia χ to Alice at the same time, χ and χ are given by | ib | ia | ib

χ = (α 0 + β 1 )a | ia | i | i (3) χ = (γ 0 + δ 1 ) | ib | i | i b 2 2 where α, β, γ and δ are complex probability amplitudes satisfying α 2 + β = 1, γ + δ 2 = 1. | | | | Without loss of generality, Alice and Bob previously share a six-particle GHZ-GHZ state, given by E E C = ψ1 ψ1 | i h A1B1A2 ⊗ B2A3B3 i (4) = 1 ( 000000 + 000111 + 111000 + 111111 ) 2 | i | i | i | i A1B1A2B2A3B3 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 13

χα=( 0+ β 1 ) a a (3) χγ=( 0+ δ 1 ) b b 22 where α , β , γ and δ are complex probability amplitudes satisfying αβ+=1 , 22 γδ+=1. Without loss of generality, Alice and Bob previously share a six-particle GHZ-GHZ state, given by C =⊗ψψ11 Appl. Sci. 2020, 10, 5500 ABA112 B 2 AB 3 3 3 of 11 1 (4) =+++() 000000 000111 111000 111111 2 ABAB112 2 AB 3 3 where the particles A1, A2 and A3 belong to Alice and particles B1, B2 and B3 belong to Bob, as shown where the particles A1 , A2 and A3 belong to Alice and particles B1 , B2 and B3 belong to Bob, in Figure1.as shown in Figure 1.

A1 B1

a A2 B2

A3 B3

Figure 1. TheFigure composite 1. The composite Greenberger–Horne–Zeilinger Greenberger–Horne–Zeilinger (GHZ)-GHZ (GHZ)-GHZ state previously previously shared shared between Alice and Bob. between Alice and Bob. Υ The stateThe of thestate initial of the initial eight-particle eight-particle system system Y 1 is is given given by | 1i Υ =⊗⊗⊗ψψ11 χχ 1 ABAB ABab E a b E 112 2 33 ABA112 B 2 AB 331 1 Y = ψ ψ χ χ 1 A1B1A2B2A3B3ab A B A B A B a b (5) h =+|1 i ⊗+⊗+⊗+1 1 2 ⊗ i αβ2 3 3 ⊗ | i ⊗ γδ| i (5) 1 () 000 111() 000 111 ( 0 1 )ab ( 0 1 ) = ( 000 +2 111 ) ABA112( 000 + 111 ) B 2 AB 33 (α 0 + β 1 ) (γ 0 + δ 1 ) 2 | i | i A1B1A2 ⊗ | i | i B2A3B3 ⊗ | i | i a ⊗ | i | i b As shown in Figure 2, to achieve bidirectional qubit transmission, Alice and Bob perform As shownControl-NOT in Figure (CNOT)2, togates achieve and Hadamard bidirectional gates in qubitparallel, transmission, and the state of Alice the entire and system Bob perform Control-NOTbecomes (CNOT) gates and Hadamard gates in parallel, and the state of the entire system becomes 1 Υ' =++−{[n 000 (αβ 0 1 ) 100 ( αβ 0 1 ) 1 1 aA ABB11 aA A Y01 = 4 [ 000 aA12A (α 0 + β 1 ) + 100 12 aA A (α 0 β 1 ) | i 4 | i 1 2 | i | i B1 | i 1 2 | i − | i B1 +++− 011 (αβ 1 0 ) 111 ( αβ 1 0 ) ] + 011 aA AA (α 1 + β 0BB11) + 111 aA A aA A (α 1 β 0 ) ] | i 121 2 | i | i B1 | 12i 1 2 | i − | i B1 (6) (6) ⊗++−[ 000 γδ(γ 0 + δ 1 ) + 100 γδ(γ 0 δ 1 ) [ 000bB2B (3 0 1 )AAA3 100 (bB2 0B3 1 ) A3 ⊗ | ibB23 B| i | i33| bB 23 B i | i − | i o + 011 (γ 1 + δ 0 ) + 111 (γ 1 δ 0 ) ] +++−bB2B3γδA3 bB γδ2B3 A3 011 ( 1 0 )AA 111 ( 1 0 ) ]} | ibB23 B| i | i33| bB 23 B i | i − | i Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13

χ a a

A1 A 2 χ b A3 B 1 χ a B2 B χ 3 b

Figure 2. QuantumFigure 2. Quantum circuit forcircuit one-hop for one-hop bidirectional bidirectional quantum quantum communication,communication, where where the dashed the dashed lines lines represent classical channels and the solid lines represent quantum channels. represent classical channels and the solid lines represent quantum channels. After that, Alice and Bob perform quantum measurements (denoted as PM in Figure 2) on the After that, Alice and Bob perform quantum measurements (denoted as PM in Figure2) on the particles a , A1 , A2 and b , B2 , B3 under the basis {0， 1}, and then inform each other of particles a, A , A and b, B , B under the basis 0 , 1 , and then inform each other of their measurement their1 measurement2 2 results3 through classical{| i chan| i}nel. According to the received measurement results through classical channel. According to the received measurement results, Alice and Bob results, Alice and Bob perform appropriate unitary operations on the particle A3 and particle B1 perform appropriate unitary operations on the particle A and particle B to recover the original to recover the original qubit. For example, when Alice’s measurement3 result is 0001 and Bob’s aA12 A qubit. For example, when Alice’s measurement result is 000 aA1A2 and Bob’s measurement result measurement result is 011 , from Equation (6) we can| seei that the quantum state of the is 011 , from Equation (6) webB23 can B see that the quantum state of the particles A and B would | ibB2B3 3 1 particles A and B would be (0αβ+⊗+ 1)(1 γδ 0). Alice can retrieve the original 3 1 BA13

qubit by performing the identity operation I on particle B1 and Bob can retrieve the original qubit

by performing the Pauli-X operation on particle A3 , respectively. Table 1 lists Alice’s and Bob’s unitary operations corresponding to the received measurement results.

Table 1. The relations between Alice’s and Bob’s measurement results and unitary operations.

Received Measurement Result Unitary Operation 000 I

011 X

100 Z

111 ZX

Table 1. The relations between Alice’s and Bob’s measurement results and unitary operations.

Received Measurement Result Unitary Operation 000 I | i 011 X | i 100 Z | i 111 ZX | i

Similarly, when other types of GHZ-GHZ states are previously shared between Alice and Bob, the unitary operations corresponding to Alice’s and Bob’s measurement results are listed in Table2. The ﬁrst rowAppl. shows Sci. 2020 the, 10, x GHZ-GHZ FOR PEER REVIEW states shared between Alice and Bob, while the ﬁrst column5 of 13 shows Alice’s and Bob’s measurement results. Table 2. The unitary operations to realize one-hop bidirectional quantum communication. Table 2. The unitary operations to realize one-hop bidirectional quantum communication. ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ABA ABA ABA ABA ABA ABA ABA ABA 112 112 112 112 112 112 112 112 |ψ1> |ψψ21> ψ 2|ψ3> ψ 3 |ψ4 > ψ 4 |ψ5ψ>5 ψ|ψ6 6> ψ 7 |ψ7> ψ 8 |ψ 8> A1B1A2 BA233AB1B1A2 B233ABA1B1A2 B233AB A1B1AB2332AB A1BB233AB1A2 B233ABA1B1A2 B233AB A1B1A2B233AB A1B1A2 |ψ1> |ψ2> |ψ3> |ψ4> |ψ5> |ψ6> |ψ7> |ψ8> B0002A3B3 IB 2A3B3 B2A3B3 X B2A3B 3 B Z2A 3B3 B2A3B3 ZX B2A3B3 B2A3B3 000 I X Z ZX | i 011 X I ZX Z 011 X I ZX Z | i 100 Z ZX I X 100 Z ZX I X | i 111 ZX Z X I 111 ZX Z X I | i 001 I X Z ZX 001 I X Z ZX | i 010 X I ZX Z 010 X I ZX Z | i 101 Z ZX I X 101 Z ZX I X | i 110 ZX Z X I 110 ZX Z X I | i 3. Two-Hop Directional Quantum Communication 3. Two-Hop Directional Quantum Communication χα=( 0+ β 1 ) Assume that Alice intends to send an arbitrary qubit a a to Bob; at the = ( + ) 22 Assumesame that time, Alice Bob intends intends to sendto send an arbitraryχγ=( 0 qubit+ δ 1 )χ ato Alice,α 0 whereβ 1 aαβto Bob;+= at1 the and same time, b | bi | i | i 2 2 2 2 Bob intends to22 send χ b= (γ 0 + δ 1 )b to Alice, where α + β = 1 and γ + δ = 1, while there γδ+=| 1i, while |therei is |noi GHZ state shared directly| | between Alice and Bob. |Suppose| that is no GHZ state shared directly between Alice and Bob. Suppose that Candy previously shared a Candy previously shared a six-particle GHZ-GHZ state with Alice and Bob, respectively, as shown six-particlein GHZ-GHZ Figure 3. state with Alice and Bob, respectively, as shown in Figure3.

B A1 C1 C4 1

C C B2 a A2 2 5 C B A3 C3 6 3

Figure 3. TheFigure composite 3. The composite GHZ-GHZ GHZ-GHZ entanglement entanglement channel channel in in a two-hopa two-hop quantumquantum communication communication case. case.

Without loss of generality, assume that ψψ11− was previously shared between A112CA C 23 AC 3 Alice and Candy, while ψψ44− was previously shared between Candy and Bob. CBC415 BCB 2 63

The state of the initial fourteen-particle system Υ2 is given by Appl. Sci. 2020, 10, 5500 5 of 11

E E Without loss of generality, assume that ψ1 ψ1 was previously shared between E E A1C1A2 − C2A3C3 Alice and Candy, while ψ4 ψ4 was previously shared between Candy and Bob. The state C4B1C5 − B2C6B3 of the initial fourteen-particle system Y is given by | 2i E E E E 1 1 4 4 Y2 = ψ ψ ψ ψ χ a χ b | i A1C1Ah 2 ⊗ C2A3C3 ⊗ C4B1C5 ⊗ B2C6B3 ⊗ | ii ⊗ | i = 1 ( 000 + 111 ) ( 000 + 111 ) 2h | i | i A1C1A2 ⊗ | i | i C2A3C3i (7) 1 ( 011 + 100 ) ( 011 + 100 ) ⊗ 2 | i | i C4B1C5 ⊗ | i | i B2C6B3 (α 0 + β 1 ) (γ 0 + δ 1 ) ⊗ | i | i a ⊗ | i | i b As shown in Figure4, to achieve bidirectional two-hop qubit transmission, Alice, Bob, and Candy perform CNOT, Hadamard operations, and quantum measurements in parallel. After that, the measurement results of Alice and Candy and the type of the composite GHZ-GHZ channel between Alice and Candy are transmitted to Bob. Similarly, the measurement results of Bob and Candy and the type of the composite GHZ-GHZ channel between Bob and Candy are transmitted to Alice. Based on the received information, Alice and Bob perform appropriate unitary operations on the particles A3 and B1 to recover the original qubit.

CNOT Y2 | i → 1 Y02 = αγ 00 [ 000101011011 + 000000011100 + 000101100011 + 000000100100 | i 4 | iab | i | i | i | i + 000010011011 + 000111011100 + 000010100011 + 000111100100 | i | i | i | i + 111101110011 + 111000110100 + 111101001011 + 111000001100 | i | i | i | i + 111010110011 + 111111110100 + 111010001011 + 111111001100 ]A C A C A C C B C B C B | i | i | i | i 1 1 2 2 3 3 4 1 5 2 6 3 + 1 αδ 01 [ 000101011110 + 000000011001 + 000101100110 + 000000100001 4 | iab | i | i | i | i + 000010011110 + 000111011001 + 000010100110 + 000111100001 | i | i | i | i + 111101110110 + 111000110001 + 111101001110 + 111000001001 | i | i | i | i + 111010110110 + 111111110001 + 111010001110 + 111111001001 ]A C A C A C C B C B C B | i | i | i | i 1 1 2 2 3 3 4 1 5 2 6 3 + 1 βγ 10 [ 101101011011 + 101000011100 + 101101100011 + 101000100100 4 | iab | i | i | i | i + 101010011011 + 101111011100 + 101010100011 + 101111100100 | i | i | i | i + 010101110011 + 010000110100 + 010101001011 + 010000001100 | i | i | i | i + 010010110011 + 010111110100 + 010010001011 + 010111001100 ]A C A C A C C B C B C B | i | i | i | i 1 1 2 2 3 3 4 1 5 2 6 3 + 1 βδ 11 [ 101101011110 + 101000011001 + 101101100110 + 101000100001 4 | iab | i | i | i | i + 101010011110 + 101111011001 + 101010100110 + 101111100001 | i | i | i | i + 010101110110 + 010000110001 + 010101001110 + 010000001001 (8) | i | i | i | i + 010010110110 + 010111110001 + 010010001110 + 010111001001 ]A C A C A C C B C B C B | i | i | i | i 1 1 2 2 3 3 4 1 5 2 6 3 Hadamard → = 1 αγ[( 0 + 1 ) ( 0 + 1 ) ] 16 | i | i a | i | i b ⊗ [ 000101011001 + 010101011001 000101011011 010101011011 + ] | i | i − | i − | i ··· A1C1A2C2A3C3C4B1C5B2C6B3 + 1 αδ[( 0 + 1 ) ( 0 1 ) ] 16 | i | i a | i − | i b ⊗ [ 000111011001 + 000111011011 + 010111011001 + 010111011011 + ...] | i | i | i | i A1C1A2C2A3C3C4B1C5B2C6B3 + 1 βγ[( 0 1 ) ( 0 + 1 ) ] 16 | i − | i a | i | i b ⊗ [ 000101001001 010101001001 000101001011 + 010101001011 + ...] | i − | i − | i | i A1C1A2C2A3C3C4B1C5B2C6B3 + 1 βδ[( 0 1 ) ( 0 1 ) ] 16 | i − | i a | i − | i b ⊗ [ 000111001001 010111001001 + 000111001011 010111001011 + ...] | i − | i | i − | i A1C1A2C2A3C3C4B1C5B2C6B3 = [ 000011101001 abA A C C C C C C B B + ... +] [(γ 0 + δ 1 ) (α 1 β 0 ) ] | i 1 2 1 2 3 4 5 6 2 3 ⊗ | i | i A3 ⊗ | i − | i B1 +[ 010011101001 abA A C C C C C C B B + ... +] [(γ 0 δ 1 ) (α 1 β 0 ) ] | i 1 2 1 2 3 4 5 6 2 3 ⊗ | i − | i A3 ⊗ | i − | i B1 +[ 100011101001 abA A C C C C C C B B + ... +] [(γ 0 + δ 1 ) (α 1 + β 0 ) ] | i 1 2 1 2 3 4 5 6 2 3 ⊗ | i | i A3 ⊗ | i | i B1 +[ 110011101001 abA A C C C C C C B B + ... +] [(γ 0 δ 1 ) (α 1 + β 0 ) ] | i 1 2 1 2 3 4 5 6 2 3 ⊗ | i − | i A3 ⊗ | i | i B1 + ... Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 13

Υ =⊗⊗⊗ψψψ11 4 ψ 4 ⊗⊗ χχ 2 ab AC112 A C 23 AC 3 C 415 BC B 2 C 63 B 1 =+() 000 111 ⊗+() 000 111 AC112 A C 233 AC 2 (7) 1 ⊗+() 011 100 ⊗+() 011 100 2 CBC415 BCB 2 63 ⊗+αβ ⊗+ γδ ( 0 1 )ab ( 0 1 )

As shown in Figure 4, to achieve bidirectional two-hop qubit transmission, Alice, Bob, and Candy perform CNOT, Hadamard operations, and quantum measurements in parallel. After that, the measurement results of Alice and Candy and the type of the composite GHZ-GHZ channel between Alice and Candy are transmitted to Bob. Similarly, the measurement results of Bob and Candy and the type of the composite GHZ-GHZ channel between Bob and Candy are transmitted to Alice. Based on the received information, Alice and Bob perform appropriate unitary operations on Appl. Sci. 2020, 10, 5500 6 of 11 the particles A3 and B1 to recover the original qubit.

χ a a

A1 A χ 2 b A3

C2 C3

C4 C5

C1 χ a B1

B2 B χ 3 b FigureFigure 4.4. QuantumQuantum circuitcircuit forfor two-hoptwo-hop bidirectionalbidirectional quantumquantum communication.communication.

For example, if Bob receives measurement results 000 and 101 , and Alice receives | iaA1A2 | iC1C4C5 measurement results 011 and 101 , the quantum state of the particles A and B would | iC6C2C3 | ibB2B3 3 1 be (γ 0 δ 1 ) (α 1 β 0 ) . Alice can retrieve the original qubit by performing the Pauli-XZ | i − | i A3 ⊗ | i − | i B1 operation on particle B1, and Bob can retrieve the original qubit by performing the Pauli-Z operation on particle A3. If the above two-hop qubit transmission is realized hop-by-hop, after Alice performs CNOT, Hadamard operations, and the projective measurement, the quantum state of Candy’s particle would be I χ when she was informed of Alice’s measurement result of 000 , according to Table2. After | ia | iaA1A2 Candy performs CNOT, Hadamard operations, and the projective measurement, the quantum state of Bob’s particle would be ZX(I χ ) = ZX χ when he was informed of Candy’s measurement result of | ia | ia 101 . Similarly, in the other direction, when Bob’s measurement result is 101 , the quantum | iC1C4C5 | ibB2B3 state of Candy’s particle would be ZX χ . If Candy’s measurement result is 011 6 2 3 , the quantum | ib | iC C C state of Alice’s particle would be X(ZX χ ) = Z χ , which is compatible with Equation (8). | ib | ib 4. Multi-Hop Bidirectional Quantum Communication In this section, we generalize two-hop bidirectional quantum communication to the multi-hop case.

4.1. N-Hop Bidirectional Quantum Communication

Speciﬁcally, in the n-hop case, the entanglement channel between the node N1 and the nodes Nn+1 is shown in Figure5. Assume that the node N intends to transmit an arbitrary qubit χ = α 0 + β 1 1 | ia | ia | ia to the node Nn+1, while the node Nn+1 intends to transmit χ b = γ 0 b + δ 1 b to the node N1 at the 2 2 | i | i | i same time, where α 2 + β = 1 and γ + δ 2 = 1. | | | | To achieve bidirectional n-hop qubit transmission, each node on the quantum path performs quantum measurements simultaneously. After that, the node N1 and all intermediate nodes send their measurement results together with the types of the composite GHZ-GHZ states shared between themselves and the next-hop nodes to the node Nn+1 through classical channels independently. Similarly, the node Nn+1 and all intermediate nodes send their measurement results together with the types of the GHZ-GHZ states to the node N1 independently. Based on the received information, the node N1 and the node Nn+1 perform appropriate unitary operations locally to recover the original qubit. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13

If the above two-hop qubit transmission is realized hop-by-hop, after Alice performs CNOT, Hadamard operations, and the projective measurement, the quantum state of Candy’s particle would be I χ when she was informed of Alice’s measurement result of 000 , according to a aA12 A Table 2. After Candy performs CNOT, Hadamard operations, and the projective measurement, the ZX() Iχχ= ZX quantum state of Bob’s particle would be aa when he was informed of Candy’s measurement result of 101 . Similarly, in the other direction, when Bob’s CC145 C measurement result is 101 , the quantum state of Candy’s particle would be ZX χ . If bB23 B b 011 Candy’s measurement result is CCC623, the quantum state of Alice’s particle would be XZX()χχ= Z bb, which is compatible with Equation (8).

4. Multi-Hop Bidirectional Quantum Communication In this section, we generalize two-hop bidirectional quantum communication to the multi-hop case.

4.1. N-Hop Bidirectional Quantum Communication

Specifically, in the n-hop case, the entanglement channel between the node N1 and the nodes

Nn+1 is shown in Figure 5. Assume that the node N1 intends to transmit an arbitrary qubit χα=0+ β 1 aaa to the node Nn+1 , while the node Nn+1 intends to transmit Appl. Sci. 2020, 10, 5500 7 of 11 χγ=0+ δ 1 αβ22+=1 γδ22+=1 bbb to the node N1 at the same time, where and .

1 1 4 1 4 1 N1 N2 N N N N Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 n n n+1 9 of 13 2 N2 5 2 5 2 a N1 2 N2 Nn Nn Nn+1 CNOT, Hadamard operations, and quantum projective measurement, the quantum state of N2 ’s 3 3 6 3 6 3 N1 T χ N2 =  particle would be 1 a , whereN 2Tii (1,2,,) n is a unitary matrix.Nn AfterNn N2 performsNn+1 CNOT,

Hadamard operations, and quantum projective measurement, the quantum state of N3 ’s particle would be TT()Nχ1 . Similarly, after all the intermediate nodes perform CNOT, NHadamardn+1 21 a N N 2 n operations, and quantum projective measurements, the quantum state of the node Nn+1 ’s particle FigureFigure 5.5. TheThe composite composite GHZ-GHZ GHZ-GHZ entanglement entanglement channel channel in thein nthe-hop n-hop quantum quantum communication communication case. would be TTT χ . case. n 21 a According to the analysis in Sectionχ3, the unitary operations of N1 and Nn+1 can be determined in To recover the original qubit locally, the node Nn+1 needs to perform the unitary a hop-by-hop manner. We take the qubit transmissiona from the node N1 to the node Nn+1 as an example. To achieve bidirectional n-hop1 qubit transmission, each node on the quantum path performs Givenoperation that nodeU onN his1 intends particle to transmitNn+1 , givenχ a byto the node Nn+1, after N1 ’s CNOT, Hadamard operations, quantum measurements simultaneously.| i After that, the node N1 and all intermediate nodes send and quantum projective measurement, the quantum state of N2’s particle would be T1 χ a, where their measurement results together with the typesn of the composite GHZ-GHZ states| i shared Ti(i = 1, 2, , n) is a unitary matrix. After N2 performs= CNOT, Hadamard operations, and quantum ··· UT∏ i (9) projectivebetween themselves measurement, and the the quantum next-hop state nodes of N ’sto particle the node would N ben+1 T through(T χ ) .classical Similarly, channels after all 3 i=1 2 1| ia theindependently. intermediate nodesSimilarly, perform the CNOT,node N Hadamardn+1 and all operations, intermediate and quantum nodes send projective their measurements,measurement where Ti(1,2,,)=  n is determined according to Table 2, based on the measurement results of the quantumi state of the node Nn+1’s particle would be Tn T2T1 χ a. results together with the types of the GHZ-GHZ states to the··· node| iN1 independently. Based on the the nodeTo recover Ni and the originalthe composite qubit χ GHZ-GHZa locally, the state node previouslyNn+1 needs shared to perform between the unitary the node operation Ni andU received information,1 the node | Ni1 and the node Nn+1 perform appropriate unitary operations on his particle N + , given by Ni+1 . n 1 locally to recover the original qubit. Yn Similarly, for the other direction of qubitU =transmission,T the node N can perform the unitary(9) According to the analysis in Section 3, the unitaryi operations 1of N1 and Nn+1 can be V N 3 i=1 χ operationdetermined in ona hop-by-hop its particle manner.1 to recover We take the the original qubit transmissionqubit b , from the node N1 to the node where Ti(i = 1, 2, , n) is determined according to Table2, based on the measurement results of the N as an example.··· Given that node N intends1 to transmit χ to the node N , after N ’s noden+1 N and the composite GHZ-GHZ state1 previously shared betweena the node N andn+1 N . 1 i VC= i i+1 Similarly, for the other direction of qubit transmission,∏ j+1 the node N can perform the unitary(10) = 1 3 jn operation V on its particle N1 to recover the original qubit χ b, = | i where Cjj+1(1,2,,) n is determined according to Table 2, based on the measurement results Y1 of the node N j+1 and the composite GHZ-GHZV = stateCj +previously1 shared between the node N j (10)and j=n N j+1 . where C + (j = 1, 2, , n) is determined according to Table2, based on the measurement results of j 1 ··· the4.2. nodeExampleNj+ of1 andMulti-Hop the composite Bidirectional GHZ-GHZ Quantum state Communication previously shared between the node Nj and Nj+1. An example is presented to better illustrate the whole process of multi-hop bidirectional 4.2. Example of Multi-Hop Bidirectional Quantum Communication quantum communication. Assume that two nodes named N1 and N5 intend to transmit two An exampleχ is presentedχ to better illustrate the whole process of multi-hop bidirectional quantum qubits and to each other. We might as well suppose that three intermediate nodes N2 , communication.a Assumeb that two nodes named N1 and N5 intend to transmit two qubits χ a and χ b →→→→ | i | i toN each3 and other. N4 Weare might involved as well in supposethe quantum that three path intermediate ‘ NNNNN12345 nodes N2, N3 and N4 ’,are as involved shown in the quantum path ‘N N N N N ’, as shown in Figure6. Figure 6. 1 → 2 → 3 → 4 → 5 χ χ b a 88 ψψ11− ψψ55− ψψ44− ψψ−

N N N N N 1 2 3 4 5 Figure 6. An example of multi-hop bidirectional quantum communication.

ψψ11− Assume that the composite GHZ-GHZ states 11 2 2 3 3 , N121NN NNN 21 2 ψψ55− ψψ44− ψψ88− 415 263, 415 263, and 415 263 were previously N232NN NNN 323 N343NN NNN 434 N454NN NNN 545 shared on the quantum path. To implement the qubit transmission from N1 to N5 , N1 and all the intermediate nodes perform quantum measurements simultaneously, then they send their Appl. Sci. 2020, 10, 5500 8 of 11

E E E E Assume that the composite GHZ-GHZ states ψ1 ψ1 , ψ5 ψ5 , N1N1N2 − N2N3N3 N4N1N5 − N2N6N3 E E E E 1 2 1 2 1 2 2 3 2 3 2 3 ψ4 ψ4 , and ψ8 ψ8 were previously shared on the quantum path. N4N1N5 N2N6N3 N4N1N5 N2N6N3 3 4 3 − 4 3 4 4 5 4 − 5 4 5 To implement the qubit transmission from N1 to N5, N1 and all the intermediate nodes perform quantum measurements simultaneously, then they send their measurement results and the type of the composite GHZ-GHZ states previously shared between themselves and the next-hop nodes to the node N5 through classical channels. Assume the node N5 receives a group of measurement results 011 sN1N1 , 100 N1N4N5 , 001 N1N4N5 , and 001 N1N4N5 from N1, N2, N3, N4 respectively; the unitary | i 1 2 | i 2 2 2 | i 3 3 3 | i 4 4 4 operation required for recovering the original qubit locally is given by

Y4 U = T = X I X ZX = ZX (11) i · · · i=1 where T1 = X, T2 = I, T3 = X, and T4 = ZX are determined according to Table2. When it comes to the qubit transmission from N5 to N1, N5 and all the intermediate nodes perform quantum measurements simultaneously, then they send their measurement results and the type of the composite GHZ-GHZ state previously shared between themselves and the next-hop node to the node N1 through classical channels. Assume the node N1 receives a group of measurement results 011 N6N2N3 , 100 N6N2N3 , 101 N6N2N3 , and 110 dN2N3 from N2, N3, N4, and N5 respectively; the unitary | i 2 2 2 | i 3 3 3 | i 4 4 4 | i 5 5 operation required for recovering the original qubit locally is given by

Y1 V = C + = I ZX I X = Z (12) j 1 · · · j=4 where C5 = I, C4 = ZX, C3 = I, and C2 = X are determined according to Table2.

5. Discussions and Conclusions

5.1. Discussions In the process of multi-hop quantum communication, the quantum communication delay is introduced mostly by quantum measurements, unitary operations, and transmission of classical information. Due to the limited decoherence time in the quantum memory and the demand for QOS (Quality of Service), a short delay is expected in quantum communication. Assume that each quantum measurement takes dm milliseconds, unitary operation takes du milliseconds, and one-hop transmission of classic information through classical channel takes dt milliseconds. In our scheme, as shown in Figure7, quantum measurements (CNOT, Hadamard operations, and projective measurements are included) are performed simultaneously, and classical information d( = ) is sent to the destination node independently. Taking Hi i 1, ... , n to denote the hop count of the classical information transmitted from the i-th node to the destination node, therefore the total end-to-end quantum communication delay in the n-hop case is given by

n do d = max H dt + du + dm (13) A,total i × For hop-by-hop qubit transmission [7], as shown in Figure8, the total end-to-end quantum communication delay is given by dB,total = n(dt + du + dm) (14)

Given that dt = 2du = 2dm = 0.6ms, the end-to-end quantum communication delay in our scheme and hop-by-hop qubit transmission scheme is demonstrated by Figure9. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13

Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13

Appl. Sci. 2020, 10, 5500 9 of 11 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13 N1 N2 Nn Nn+1

N1 N2 Nn Nn+1

Figure 7. The proposed scheme in this paper.

For hop-by-hop qubit transmission [7], as shown in Figure 8, the total end-to-end quantum communication delay is given by N1 N2 Nn Nn+1 Figure 7. The proposed=++ scheme in this paper. dndddB,total() t u m (14) For hop-by-hop qubit transmission [7], as shown in Figure 8, the total end-to-end quantum communication delay is given by Nn+ =++ dndddB,total() t u m (14) FigureFigure 7.7. TheThe proposedproposed schemescheme inin thisthis paper.paper.

For hop-by-hop qubit transmission [7], as shown in Figure 8, the total end-to-end quantumNn+ N1 N2 Nn Nn+1 communication delay is given by Figure 8. Hop-by-hop=++ qubit transmission. dndddB,total() t u m (14) === Given that dddtum220.6 ms, the end-to-end quantum communication delay in our N1 N2 Nn Nn+1 scheme and hop-by-hop qubit transmission scheme is demonstrated by Figure 9. Nn+ Figure 8. Hop-by-hop qubit transmission.

=== Given that dddtum220.6 ms, the end-to-end quantum communication delay in our scheme and hop-by-hop qubit transmission scheme is demonstrated by Figure 9. N1 N2 Nn Nn+1

Figure 8. Hop-by-hop qubit transmission.

=== Given that dddtum220.6 ms, the end-to-end quantum communication delay in our scheme and hop-by-hop qubit transmission scheme is demonstrated by Figure 9.

FigureFigure 9. 9.The The comparison comparison of of two two schemes schemes in in terms terms of of communication communication delay. delay.

From Figure9, it is obvious that our scheme has a shorter end-to-end quantum communication delay. In a wide-area quantum communication network where the hop count number n is large, our scheme shows obvious advantage in the end-to-end quantum communication delay. Figure 9. The comparison of two schemes in terms of communication delay. In addition to the shorter quantum communication delay, our scheme shows other notable advantages, such as: (i) Duplex quantum communication is available for two arbitrary nodes compared with other quantum network schemes [6–8,11,12]. (ii) The composite GHZ-GHZ states are used as the quantum channel, which is less prone to the decoherence eﬀects caused by the surrounding

Figure 9. The comparison of two schemes in terms of communication delay. Appl. Sci. 2020, 10, 5500 10 of 11 environment [22]. (iii) Only quantum projective measurements and unitary operations are needed to implement qubit transmission. Auxiliary qubits are not required, in contrast to the bidirectional quantum communication scheme based on GHZ-Bell states [20]. (iv) In our scheme, for an intermediate node, it is not necessary to recover the original qubit locally and then teleport it to its next-hop node. Therefore, information leakage can be avoided compared with the hop-by-hop qubit transmission scheme [7].

5.2. Conclusions In summary, we propose a bidirectional quantum communication protocol using the composite GHZ-GHZ state as quantum channel. Two arbitrary nodes in a quantum communication network can transmit qubits to each other based on quantum measurements performed in parallel by all the nodes involved. The original qubits can be retrieved locally at the destination site based on the measurement results and the types of GHZ channel. Information leakage is avoided in our scheme, and it has a shorter delay for the end-to-end quantum communication, in contrast to the hop-by-hop qubit transmission scheme. There are many avenues for future work and extensions. One aspect is the imperfection of quantum channel, which is crucial for the eﬃciency and ﬁdelity of qubit transmission. Another important aspect is the extension to multi-user quantum communication, which is a realistic scenario for future quantum networks.

Author Contributions: Data curation, S.S. and N.C.; methodology, S.S. and N.C.; supervision, B.Y.; validation, S.S.and N.C.; visualization, B.Y.; writing—original draft, S.S.; writing—review and editing, N.C. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 61701285). Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Bennett, C.H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Wootters, W.K. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett. 1993, 70, 1895–1899. [CrossRef] [PubMed] 2. Espoukeh, P.; Pedram, P. Quantum teleportation through noisy channels with multi-qubit GHZ states. Quant. Inf. Process. 2014, 13, 1789–1811. [CrossRef] 3. Joo, J.; Park, Y.J.; Oh, S.; Kim, J. Quantum teleportation via a W state. New J. Phys. 2003, 5, 136. [CrossRef] 4. Karlsson, A.; Bourennane, M. Quantum teleportation using three-particle entanglement. Phys. Rev. A 2002, 58, 4394–4400. [CrossRef] 5. Cao, Z.L.; Song, W. Teleportation of a two-particle entangled state via W class states. Physica A 2005, 347, 177–183. [CrossRef] 6. Wang, K.; Yu, X.T.; Lu, S.L.; Gong, Y.X. Quantum wireless multihop communication based on arbitrary Bell pairs and teleportation. Phys. Rev. A 2014, 89, 022329. [CrossRef] 7. Yu, X.T.; Zhang, Z.C.; Xu, J. Distributed wireless quantum communication networks with partially entangled pairs. Chin. Phys. B 2014, 23, 010303. [CrossRef] 8. Chen, N.; Quan, D.X.; Pei, C.X.; Yang, H. Quantum communication for satellite-to-ground networks with partially entangled states. Chin. Phys. B 2015, 24, 020304. [CrossRef] 9. Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 2009, 81, 865–942. [CrossRef] 10. Tashima, T.; Tame, M.S.; Ozdemir,˝ ¸S.K.;Nori, F.; Koashi, M.; Weinfurter, H. Photonic multipartite entanglement conversion using nonlocal operations. Phys. Rev. A 2016, 94, 052309. [CrossRef] 11. Zhan, H.T.; Yu, X.T.; Xiong, P.Y.; Zhang, Z.C. Multi-hop teleportation based on W state and EPR pairs. Chin. Phys. B 2016, 25, 050305. [CrossRef] 12. Zou, Z.Z.; Yu, X.T.; Gong, Y.X.; Zhang, Z.C. Multihop teleportation of two-qubit state via the composite GHZ–Bell channel. Phys. Lett. A 2017, 381, 76–81. [CrossRef] Appl. Sci. 2020, 10, 5500 11 of 11

13. Li, Y.H.; Li, X.L.; Sang, M.H.; Nie, Y.Y.; Wang, Z.S. Bidirectional controlled quantum teleportation and secure direct communication using five-qubit entangled state. Quant. Inf. Process. 2013, 12, 3835–3844. [CrossRef] 14. Hassanpour, S.; Houshmand, M. Bidirectional teleportation of a pure EPR state by using GHZ states. Quant. Inf. Process. 2016, 15, 905–912. [CrossRef] 15. Duan, Y.J.; Zha, X.W. Bidirectional quantum controlled teleportation via a six-qubit entangled state. Int. J. Theor. Phys. 2014, 53, 3780–3786. [CrossRef] 16. Zhang, D.; Zha, X.W.; Li, W.; Yu, Y. Bidirectional and asymmetric quantum controlled teleportation via maximally eight-qubit entangled state. Quant. Inf. Process. 2015, 14, 3835–3844. [CrossRef] 17. Sang, M.H. Bidirectional quantum teleportation by using five-qubit cluster state. Int. J. Theor. Phys. 2016, 55, 1333–1335. [CrossRef] 18. Fang, S.H.; Jiang, M. A novel scheme for bidirectional and hybrid quantum information transmission via a seven-qubit state. Int. J. Theor. Phys. 2017, 57, 523–532. [CrossRef] 19. Fang, S.H.; Jiang, M. Bidirectional and asymmetric controlled quantum information transmission via five-qubit brown state. Int. J. Theor. Phys. 2017, 56, 1530–1536. [CrossRef] 20. Cai, R.; Yu, X.T.; Zhang, Z.C. Bidirectional teleportation protocol in quantum wireless multi-hop network. Int. J. Theor. Phys. 2018, 57, 1723–1732. [CrossRef] 21. Zhou, R.G.; Xu, R.; Lan, H. Bidirectional quantum teleportation by using six-qubit cluster state. IEEE Access 2019, 7, 44269–44275. [CrossRef] 22. Saha, D.; Panigrahi, P.K. N-qubit quantum teleportation, information splitting and superdense coding through the composite GHZ–Bell channel. Quant. Inf. Process. 2011, 11, 615–628. [CrossRef]

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