Yiru Sun, Yuling Chen , Haseeb Ahmad and Zhanhong Wei

Abstract: In this paper, we propose an asymmetric controlled bidirectional transmission protocol. In the protocol, by using the thirteen-qubit entangled state as the quantum channel, Alice can realize the transmission of a two-qubit equatorial state for Bob and Bob can transmit a four-qubit equatorial state for Alice under the control of Charlie.Firstly, we give the construction of the quantum channel, which can be done by performing several H and CNOT operations. Secondly, through implementing the appropriate measurements and the corresponding recovery operations, the desired states can be transmitted simultaneously, securely and deterministically. Finally, we analyze the performance of the protocol, including the efficiency, the necessary operations and the classical communication costs. And then, we describe some comparisons with other protocols. Since our protocol does not require auxiliary particles and additional operations, the classic communication costs less while achieving the multi-particle bidirectional transmission, so the overall performance of the protocol is better.

Keywords: Controlled, asymmetric, bidirectional quantum state transmission, remote state preparation.

1 Introduction

By using the quantum entanglement resources, a variety of quantum communication protocols were put forward to solve the problem of quantum information transmission,such as quantum secure direct communication [Wang, Deng, Li et al. (2005)], quantum secret sharing [Lin, Guo, Xu et al. (2016); Chen, Tang, Xu et al. (2018)], quantum key management [Xu, Chen, Duo et al. (2015); Liu, Xu, Yang et al. (2018)], quantum steganography [Wei, Chen, Niu et al. (2015); Qu, Cheng, Liu et al. (2018); Qu, Chen, Ji et al (2018)], quantum teleportation (QT) [Fortes and Rigolin(2017); Bennett, Brassard,Crépeau et al. (1993); Tan, Li and Yang (2018)] and remote state preparation (RSP) [Pati(2000); Cavaillès, Le Jeannic, Raskop et al. (2017]. In a RSP, the sender (Alice) can prepare a known state for a remote receiver (Bob) via a shared quantum channel and some classical communications. So far, many different protocols were proposed, such as controlled RSP [Huang and Zhao (2017); Chen, Ma, Su et al.(2012); Liu and Hwang(2014); Wang, Zeng and Li (2015)], joint RSP [Choudhury and Dhara(2015);An(2010);Chen, Xia, Song et al.(2010)], deterministic RSP [Xu, Chen, Dou, et al. (2016); Chen, Su,Xu et al. (2014), Luo, Chen, Ma et al. (2010); Qu, Wu, Wang et al. (2017)] and lowentanglement RSP [Devetak and Berger(2001)]. In the CRSP protocol, some re-searchers also pay attention to the related research of controller’s power [Li and Shohini(2017)]while introducing the controller. After that, some variants of quantum teleportation and RSP protocols were proposed: bidirectional teleportation [Kiktenko, Popov and Fedorov(2016); Li, Li, Sang et al. (2013)], hierarchical RSP [Shukla, Thapliyal and Pathak (2017)]and controlled bidirectional RSP (CBRSP) [Cao and Nguyen(2013); Sharma, Shukla,Banerjee et al. (2015); Peng, Bai and Mo(2015); Wang and Mo(2017); Zhang, Zha, Duan et al. (2016a, 2016b); Sang (2017); Song, Ni, Wang et al. (2017); Wu, Zha and Yang(2018); Chen, Sun, Xu et al. (2017); Sang and Nie (2017); Fang and Jiang (2018); Ma,Chen, Li et al. (2017)], for example. There are also some related quantum communication protocols that have been experimentally implemented [Zhang, Goebel, Wagenknecht et al.(2006); Sisodia, Shukla, Thapliyal et al. (2017); R?dmark, Wie?niak, ?ukowski et al.(2013); Luo, Chen, Yang et al. (2012); Liu, Gao, Yu et al. (2018); Liu, Wang, Yuan et al.(2016)]. In this paper, we focus on the study of CBRSP.

Cao et al. [Cao and Nguyen(2013)] presented the first CBRSP protocol, which realizes the bidirectional transmission of the single-qubit state through some classical communications and local operations. After that, by using different entangled states as the quantum channel, many protocols [Sharma, Shukla, Banerjee et al. (2015); Peng, Bai and Mo(2015); Wang and Mo(2017); Zhang, Zha, Duan et al. (2016a, 2016b); Sang(2017); Song, Ni, Wang et al. (2017); Wu, Zha and Yang (2018)] can achieve the bidirectional transmission of single-qubit state. Sharma et al. [Sharma, Shukla, Banerjee et al. (2015)] gave three protocols, including the probabilistic, the deterministic and the joint CBRSP protocols. In Peng et al. [Peng, Bai and Mo(2015); Wang and Mo(2017)],two five-party joint CBRSP protocols were presented via the eight-and seven-qubit entangled state as the quantum channel, respectively. Furthermore, in these protocols[Cao and Nguyen(2013); Sharma, Shukla, Banerjee et al. (2015); Zhang, Zha, Duan et al.(2016a, 2016b); Sang (2017); Song, Ni, Wang et al. (2017)], the participants need the help of auxiliary particles and additional operations to complete the transmission task.Specifically, two controlled bidirectional hybrid of RSP and QT protocols were proposed by Sang [Sang (2017)] and Wu et al. [Wu, Zha and Yang (2018)], respectively. In 2017,by using thirteen-qubit entangled state as the quantum channel, a CBRSP protocol was presented by Chen et al. [Chen, Sun, Xu et al. (2017)], where Alice and Bob can transmit an arbitrary three-qubit state to each other simultaneously.

The above CBRSP protocols are symmetric. Moreover, there are many asymmetric protocols. Song et al. [Song, Ni, Wang et al. (2017)] put forward an asymmetric bidirectional RSP protocol to prepare single- and two-qubit state. An asymmetric bidirectional hybrid of RSP and QT protocol was proposed by Sang et al. [Sang and Nie(2017)], where Alice teleports a single-qubit state to Bob and Bob prepares a two-qubit state to Alice at the same time. Fang et al. [Fang and Jiang (2018)] investigated two asymmetric protocols for bidirectional hybrid of RSP and QT, where also studied bidirectional transmission of single- and two-qubit state. Further-more, in Ma et al. [Ma,Chen, Li et al. (2017)], the senders complete the bidirectional transmission of a four-qubit cluster-type state and a single-qubit state. However, in these protocols [Song, Ni, Wang et al. (2017); Fang and Jiang (2018); Ma, Chen, Li et al. (2017)], the sender or receiver require the auxiliary particles and additional operations. And the multi-qubit state prepared by protocol in Ma et al. [Ma, Chen, Li et al. (2017)] is only a special form. Chen et al. [Chen, Su, Xu et al. (2014)] investigated the quantum state secure transmission of equatorial and general state, respectively. Wei et al. [Wei, Shi, Zhu et al. (2018)]presented a protocol for remotely preparing an arbitrary n-qubit equatorial state via n two-qubit maximally entangled states as the quantum channel. Inspired by their protocols,we study the bidirectional transmission of two- and four-qubit equatorial state.

In this paper, we propose a controlled bidirectional transmission protocol. With the control of Charlie, Alice and Bob can realize the bidirectional transmission of two- and four-qubit equatorial state. First, by using several H and CNOT operations, the thirteen-qubit entangled state can be constructed as the quantum channel. Second, through carrying out the proper measurement and recovery operations, Alice and Bob can recover the prepared state simultaneously, securely and deterministically. In the end, we analyze the performance of the protocol and describe some comparisons with other protocols, including the efficiency, the necessary operations and the classical communication costs. After that,since our protocol does not require auxiliary particles and additional operations, the classic communication costs less while achieving the bidirectional transmission of two- and fourqubit equatorial state, so the protocol has the better overall performance.

The paper is structured as follows. We propose a controlled bidirectional transmission protocol in Section 2. Then, in Section 3, some discussions are given. In the end, we describe the conclusions in Section 4.

2 The controlled bidirectional quantum state transmission protocol

In this section, we describe the CBRSP protocol in detail. In the protocol, under the control of Charlie, Alice transmits a two-qubit equatorial state to Bob and Bob transmits a four-qubit equatorial state to Alice by using the thirteen-qubit entangled state as the quantum channel. We give the construction of the quantum channel at first. Then, the CBRSP protocol is described.

2.1 Quantum channel

The quantum channel is constructed according to the principles in Thapliyal et al.[Thapliyal, Verma and Pathak(2015)]. The thirteen-qubit product stateis used as an input state, where

We construct the thirteen-qubit entangled state as the quantum channel by using H and CNOT operations as follows.

Firstly, we implement an H operation on particle 1.is transformed to:

Secondly, six CNOT operations are operated on the particle pairs (1, 2), (1, 4), (1, 6), (1,8), (1, a), (1, c), where the particle 1 is used as the controlled particle and each of six particles 2, 4, 6, 8, a, c are used as the target particles.is transformed to:

Thirdly, after executing six H operations on the particles 2, 4, 6, 8, a, c, we carry out six CNOT operations on the particle pairs (2, 3), (4, 5), (6, 7), (8, 9), (a, b), (c, d), where the particles 2, 4, 6, 8, a, c are used as the controlled particles and 3, 5, 7, 9, b, d are used as the target particles, respectively.

Finally, we can construct the thirteen-qubit entangled stateas the quantum channel:

The generation and maintenance of the multi-qubit entangled state is a difficult task, but even so, some experiments results [Zhang, Goebel, Wagenknecht et al. (2006); Sisodia,Shukla, Thapliyal et al. (2017); R?dmark, Wie?niak, ?ukowski, et al. (2013); Luo, Chen,Yang et al. (2012)] have been put forward to study the multi-qubit entangled state. As a consequence, by using some advanced technology, this task can be completed no longer difficult. Moreover, the bidirectional transmission task can be performed through the controlled bidirectional teleportation and preparation protocol. It is shown that the task can be performed with only Bell states [Thapliyal and Pathak(2015)]. In this paper, we mainly research the CBRSP. That is to say, the controller is needed in our protocol. Here,we use this thirteen-qubit entangled state as the quantum channel to accomplish the bidirectional preparation of two- and four-qubit states.

2.2 The controlled bidirectional quantum state transmission protocol

In this section, inspired by protocols [Chen, Su, Xu et al. (2014); Wei, Shi, Zhu et al.(2018)], we study the bidirectional transmission of two-and four-qubit equatorial state and will describe our CBRSP protocol in detail. The process of our protocol is graphically described in Fig. 1.

In the protocol, Alice can transmit two-qubit equatorial stateto Bob and Bob can transmit four-qubit equatorial stateto Alice, where

here θi,βjare real,0≤θi, βj≤2π,i=1,2,3,j=1,2,...,15.

[A-1] Three participants pre-share the thirteen-qubit entangled state. Specifically,Charlie holds the particle 1. The particles 2, 4, 6, 8, a, c belong to Alice and the particles 3, 5, 7, 9, b, d belong to Bob.

[A-2] Alice makes two-qubit measurement on her own particles 2, 4. The measurement basisis

Bob implements four-qubit measurement on his own particles 7, 9, b, d. The measurement basis|h=0,1,2,...,15}is

We can rewrite the quantum channelas:

Where

Alice’s and Bob’s measurement results arem=0,1,2,3and h=0,1,...,15}.They will announce their measurement results through the classical channel.

[A-3] Charlie will measure his particle 1 in the basis. Then, he announces his measurement result through the classical channel.

[A-4] If Charlie’s measurement result is, Alice obtainsby performing the unitary operationsUh,6?Uh,8?Uh,a?Uh,cand Bob obtainsby carrying out the unitary operations Um,3?Um,5, where

If Charlie’s measurement result is, the unitary operations of Alice and Bob are Uh,6Zh,6?Uh,8Zh,8?Uh,aZh,8?Uh,cZh,cand Um,3Zm,3?Um,5Zm,5, respectively.

In the protocol, by using the technique of RSP, the security of quantum state transmission can be ensured. Moreover, Charlie as the controller, after Alice and Bob announcing their measurement results, measures his own particle and announces his result, which ensure the simultaneity, that is, Alice and Bob recover the desired state at the same time. The classical communication costs are 7 cbits. Furthermore, the success probability of our protocol can be calculated as

Therefore, through implementing our protocol, the desired states can be transmitted securely, simultaneously and deterministically.

Figure 1: The process of our CBRSP protocol. Each solid point represents a particle, and the solid line represents an entanglement between particles. Dotted rectangle indicates the measurement. Solid rectangle indicates the unitary operation

3 Discussions

In this section, we discuss the efficiency and the necessary operations of our proto-col at first. Furthermore, some comparisons with other protocols are given.

3.1 Efficiency

The efficiency [Yuan, Liu, Zhang et al.(2008)] is an important factor in measuring protocol performance, which can be calculated as

where qsdenotes the number of qubits that consist of the quantum information to be prepared,quis the number of the qubits that is used as the quantum channel and btis the classical bits transmitted.

Therefore, the efficiency of our protocol is calculated as

3.2 The necessary operations

In our protocol, the necessary operations only used in constructing the quantum channel are seven H operations and twelve CNOT operations. Furthermore, no additional operations are needed.

3.3 Comparisons with other protocols

In the previous protocols [Cao and Nguyen (2013); Sharma, Shukla, Banerjee et al.(2015); Peng, Bai and Mo (2015); Wang and Mo (2017); Zhang, Zha, Duan et al. (2016a,2016b); Sang (2017); Song, Ni, Wang et al. (2017); Wu, Zha and Yang (2018); Sang and Nie (2017); Fang and Jiang (2018)], they only achieved symmetric bidirectional transmission of single-qubit state or the asymmetric transmission of single-and two-qubit state. Even in Ma et al. [Ma, Chen, Li et al. (2017)], they achieved the asymmetric bidirectional preparation of single-and four-qubit cluster-type states. However, in fact,they only transmit the single-and two-qubit state. And one of the receivers needs two local auxiliary qubits and auxiliary operations to recover the four-qubit cluster-type state.The results of comparison with the pervious asymmetric protocols are given in Tab. 1.

Firstly, by using seven H operations and twelve CNOT operations, we give the construction of the quantum channel while the previous protocols have not given.Secondly, the sender or receiver of some protocols [Cao and Nguyen (2013); Sharma,Shukla, Banerjee et al. (2015); Zhang, Zha, Duan et al. (2016a, 2016b); Sang (2017);Song, Ni, Wang et al. (2017); Fang and Jiang (2018); Ma, Chen, Li et al. (2017)] needs the help of auxiliary particles and additional operations to complete the bidirectional task,which are not needed in our protocol. Thirdly, similarly, both protocols prepare fourqubit states, while the classical communication costs of our protocol are fewer than that in Ref. [Ma, Chen, Li et al. (2017)]. Last but not the least, since our protocol can realize the bidirectional transmission of two-and four-qubit equatorial states, our protocol is more efficient than other protocols.

The point-to-point quantum communication must be turned to the multi-party quantum network communication. These have a wide range of research meanings in some network structures [Guo, Zhang, Liu et al. (2017); Pang, Liu, Zhou et al. (2017); Li, Wang, Li et al. (2018); Shen, Song, Li et al. (2018)]. As regards the quantum networks, the feasibility and construction have been fully verified theoretically [Dong, Zhang, Zhang et al. (2014);Jiang, Jiang and Ling (2014); Xu, Chen, Li et al. (2015); Li, Chen, Xu et al. (2015)]. Our scheme do not need the auxiliary resources and have relatively high efficiency, so it can be easily incorporated into the design of quantum network communication.

Table 1: Comparisons with the pervious asymmetric protocols

4 Discussions

In summary, an asymmetric protocol is proposed for controlled bidirectional quantum state transmission. In this protocol, by using the thirteen-qubit entangled state as the quantum channel, we can realize the bidirectional transmission of two-and four-qubit equatorial states simultaneously, securely and deterministically. In the end, we analyze the performance of the protocol and describe some comparisons with other protocols,including the efficiency, the necessary operations and the classical communication costs.The results show that our protocol has better performance.

Acknowledgement:Project supported by NSFC (Grant Nos. U1836205, 61702040), the Major Scientific and Technological Special Project of Guizhou Province (Grant No.20183001), the Foundation of Guizhou Provincial Key Laboratory of Public Big Data(Grant No. 2018BDKFJJ016), the Foundation of State Key Laboratory of Public Big Data (Grant No. 2018BDKFJJ018) and Beijing Natural Science Foundation (Grant No.4174089).