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 1 = 1 ( 000 + 111 ), 2 = 1 ( 001 + 110 ) E p2 j i j i E p2 j i j i 3 = 1 ( 010 + 101 ), 4 = 1 ( 011 + 100 ) E p2 j i j i E p2 j i j i (1) 5 = 1 ( 000 111 ), 6 = 1 ( 001 110 ) E p2 j i − j i E p2 j i − j i 7 = 1 ( 010 101 ), 8 = 1 ( 011 100 ) p2 j i − j i p2 j i − j i These GHZ states can be transformed into each other through unitary operations, such as E E E E 2 = X X I 1 , 3 = I X I 1 E ⊗ ⊗ E E ⊗ ⊗ E 4 = X I I 1 , 5 = Z I I 1 E ⊗ ⊗ E E ⊗ ⊗ E (2) 6 = ZX X I 1 , 7 = Z X I 1 ⊗ ⊗E E ⊗ ⊗ 8 = ZX I I 1 ⊗ ⊗ ! ! ! 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 j ia χ to Alice at the same time, χ and χ are given by j ib j ia j ib χ = (α 0 + β 1 )a j ia j i j i (3) χ = (γ 0 + δ 1 ) j ib j i j i b 2 2 where α, β, γ and δ are complex probability amplitudes satisfying α 2 + β = 1, γ + δ 2 = 1. j j j j Without loss of generality, Alice and Bob previously share a six-particle GHZ-GHZ state, given by E E C = 1 1 j i h A1B1A2 ⊗ B2A3B3 i (4) = 1 ( 000000 + 000111 + 111000 + 111111 ) 2 j i j i j i j 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 j 1i Υ =⊗⊗⊗ψψ11 χχ 1 ABAB ABab E a b E 112 2 33 ABA112 B 2 AB 331 1 Y = χ χ 1 A1B1A2B2A3B3ab A B A B A B a b (5) h =+j1 i ⊗+⊗+⊗+1 1 2 ⊗ i αβ2 3 3 ⊗ j i ⊗ γδj 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 j i j i A1B1A2 ⊗ j i j i B2A3B3 ⊗ j i j i a ⊗ j i j 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 ) j i 4 j i 1 2 j i j i B1 j i 1 2 j i − j 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 ) ] j i 121 2 j i j i B1 j 12i 1 2 j i − j 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 ⊗ j ibB23 Bj i j i33j bB 23 B i j i − j i o + 011 (γ 1 + δ 0 ) + 111 (γ 1 δ 0 ) ] +++−bB2B3γδA3 bB γδ2B3 A3 011 ( 1 0 )AA 111 ( 1 0 ) ]} j ibB23 Bj i j i33j bB 23 B i j i − j i Appl.