Asymmetric controlled bidirectional remote preparation of two- and three-qubit equatorial state

In this paper, a novel asymmetric controlled bidirectional remote preparation scheme is proposed. In our scheme, Alice and Bob are not only the senders but also the receivers with the control of Charlie. By using the eleven-qubit entangled state as the quantum channel, Alice prepares an arbitrary two-qubit equatorial state for Bob and Bob prepares an arbitrary three-qubit equatorial state for Alice simultaneously. Firstly, we give the construction process of the quantum channel. Secondly, the whole recovery operations are given. Alice and Bob can recover the prepared state determinately. Thirdly, we consider the effect of the noisy environment (amplitude-damping and phase-damping) in our scheme and calculate the fidelities of the output states. Finally, since our scheme does not need additional operations and auxiliary qubits, the efficiency of our scheme is higher than that of the previous schemes.

In quantum cryptography, quantum entanglement is magic, which plays a key role in many different types of communication schemes [1][2][3] . One of the applications is the remote state preparation (RSP) [4][5][6] . In a RSP scheme, the sender can prepare a known quantum state for the remote receiver through a pre-shared quantum channel and the appropriate measurements. After that, many various schemes were presented, such as controlled RSP (CRSP) 7 , joint RSP (JRSP) 8 .
However, the previous RSP schemes are unidirectional. The first controlled bidirectional RSP (CBRSP) scheme was presented by Cao et al. 9 in 2014, where Alice and Bob send their single-qubit state to each other simultaneously. After that, in 2015, the deterministic, the probabilistic and the joint CBRSP schemes were devised by Sharma et al. 10 . And Peng et al. 11 presented a five-party joint CBRSP scheme by using eight-qubit entangled state. In 2016, utilize the non-maximally and maximally six-qubit entangled state, Zhang et al. 12,13 proposed two joint CBRSP schemes. In 2017, another five-party joint CBRSP scheme is proposed by Wang and Mo 14 by using seven-qubit entangled state as the quantum channel. Sang 15 presented a bidirectional controlled quantum information transmission scheme, where Alice teleports an arbitrary unknown single-qubit state to Bob and Bob remotely prepares an arbitrary known single-qubit state for Alice simultaneously. In 2018, a controlled bidirectional hybrid of remote state preparation and quantum teleportation scheme was proposed by Wu et al. 16 . By using thirteen-qubit entangled state as the quantum channel, Chen et al. 17 presented a symmetric CBRSP scheme, where Alice and Bob prepared an arbitrary three-qubit state to each other simultaneously. Moreover, this scheme was discussed in four types of noisy environment and no auxiliary qubits are needed.
Moreover, many asymmetric CBRSP (ACBRSP) schemes have been proposed. In 2017, an ACBRSP scheme was presented by Sang et al. 18 , where Alice teleports an arbitrary unknown single-particle state to Bob and Bob remotely prepare an arbitrary known two-qubit state to Alice. Song et al. 19 investigated an ACBRSP scheme of a single-qubit state and two-qubit state. Ma et al. 20 proposed an ACBRSP scheme of an arbitrary four-qubit cluster-type state and a single-qubit state. In 2018, a scheme for bidirectional and hybrid quantum information transmission was proposed by Fang et al. 21 , where Alice can teleport an arbitrary single-qubit state (two-qubit state) to Bob and Bob can prepare a known two-qubit state (single-qubit state) to Alice simultaneously. In the previous schemes, many of them need the help of the auxiliary qubits and additional operations [19][20][21] . Furthermore, they do not give the construction process of the quantum channel. And only in the schemes 10,17 , the noise are considered. However, noise is a necessary factor that must be considered in actual communication. In ref. 22  space can only be implemented when the dimension of the space is 2, 4 or 8, while the equatorial case can be generalized without restriction on the dimension. Although the preparation of two states to each other can be completed by using two independent RSP schemes, simultaneity and fairness cannot be guaranteed in this process. However, in our scheme, the controlled bidirectional preparation of the equatorial state, Alice and Bob can transmit the prepared state to each other simultaneously with the control of Charlie. The simultaneity and fairness can be guaranteed in our scheme. In this paper, a novel ACBRSP scheme is put forward. With the control of Charlie, Alice prepares an arbitrary two-qubit equatorial state for Bob and Bob prepares an arbitrary three-qubit equatorial state for Alice at the same time. We firstly generate the eleven-qubit entangled state as the quantum channel. Then, through the appropriate measurements and the recovery operations, Alice and Bob can reconstruct the prepared state determinately. Moreover, we consider the effect of the noisy environment in our scheme and calculate the fidelities of the output states. Last but not the least, some discussions and conclusions are given. The result shows that our scheme does not need additional operations and auxiliary qubits. And the prepared states are the arbitrary two-and three-qubit states, so the efficiency of our scheme is higher than that of the previous schemes.

Results
The construction process of the quantum channel. We use the eleven-qubit product state as an input state 23 and implement the Hadamard (H) and CNOT operations to construct the eleven-qubit entangled state as the quantum channel 24 . The process of the construction of the quantum channel is given as follows.
The input state is the eleven-qubit product state φ 1 as C2 Operate CNOT operations on the qubit pairs (1, 3), (1,5), (1,7), (1,9), (1, b), respectively, where qubit 1 is used as controlled qubit and each of five qubits 3, 5, 7, 9, b are used as target qubit. We rewrite the φ 2 as C3 After implementing the H operations on the qubits 2, 4, 6, 8, a, we execute CNOT operations on the qubit pairs (2, 3), (4,5), (6,7), (8,9), (a, b), respectively, where qubits 2, 4, 6, 8, a are used as controlled qubit and each of five qubits 3, 5, 7, 9, b is used as target qubit. The eleven-qubit entangled state can be generated as . In Fig. 1, we give the circuit diagram of the construction of the quantum channel. The ACBRSP scheme of an arbitrary two-and three-qubit equatorial state. By using eleven-qubit entangled state φ 4 as the quantum channel, we propose a novel ACBRSP scheme. In the scheme, with the control of Charlie, Alice prepares an arbitrary two-qubit equatorial state for Bob and Bob prepares an arbitrary three-qubit equatorial state for Alice. They pre-share this quantum channel at first which is constructed by Charlie. Then, Alice and Bob can complete the preparation task through the appropriate measurements and the corresponding recovery operations. In the scheme, as an honest controller, Charlie constructs the eleven-qubit entangled state as the quantum channel and holds qubit 1 in his hand. Then, he sends the qubits 2, 4, 6, 8, a to Alice and sends the qubits 3, 5, 7, 9, b to Bob. Before distributing these qubits, the participants have to check eavesdropping by using the decoy qubits to ensure the security of the distribution. The process of checking eavesdropping is the same as that in ref. 17 . After successfully passing the check eavesdropping, the quantum channel can be securely shared by Alice, Bob and Charlie. Next, we will describe our ACBRSP scheme in detail.
A1 They pre-share the quantum channel φ 4 . Alice wants to prepare a two-qubit equatorial state ϕ A for Bob and Bob wants to prepare a three-qubit equatorial state ϕ B for Alice, where Here, θ θ θ δ , , , k .. k 1, 2, , 7. A2 Alice carries out two-qubit measurement on her qubits 2, 4. The measurement basis is ( 000 , 001 , 010 , 011 , 100 , 101 , 110 , 111 )  The quantum channel φ 4 becomes:    A4 In the end, after receiving others' measurement results, Alice can recover the three-qubit equatorial state ϕ B by applying the corresponding recovery operations on her qubits 6, 8, a and Bob can recover the two-qubit equatorial state ϕ A by applying the corresponding recovery operations on his qubits 3, 5. The process of the scheme is given in Fig. 2 ( ⊗ X X 3 5 ), respectively. For other measurement results, Alice and Bob can also use corresponding recovery operation to get the prepared state. All the measurement results and corresponding recovery operations are given in Supplementary information. Furthermore, the scheme is completed deterministically.  The ACBRSP scheme in amplitude-damping and phase-damping noisy environment. It is impossible to have a noiseless communication environment in practice. For example, Sharma et al. 10 discussed the effects of two well-known noise processes, the amplitude-damping and phase-damping noise, in their scheme. In this section, we will consider the ACBRSP scheme in two types of noisy environment (amplitude-damping and phase-damping).
In ACBRSP scheme, Charlie constructs the quantum channel φ 4 and holds the qubit 1. Since the qubit 1 is not transmitted in the noisy environment, so it is not affected by the noise. Charlie sends the qubits 2, 4, 6, 8, a (3, 5, 7, 9, b) to Alice (Bob) via the channel-I (channel-II). We assume that the channel-I and channel-II are in the same noisy environment. Therefore, the qubits 2, 4, 6, 8, a (3, 5, 7, 9, b) are affected by the same Kraus operator. The density matrix of the quantum channel φ 4 can be calculated as ρ φ φ = 4 4 .
The Kraus operators. In the amplitude-damping noisy environment, the Kraus operators 25 is the decoherence rate. It is the probability of missing a photon. Because of the interaction with this noisy environment, a system undergoes energy dissipation.
In phase-damping noisy environment, it describes the loss of information about the relative phase in the quantum state. The Kraus operators 25 is the decoherence rate.
The output state. The qubits 2, 4, 6, 8, a (3, 5, 7, 9, b) are transmitted in the amplitude-damping and phase-damping noisy environment, we rewrite the density matrix ρ as ε ρ ( )   The fidelity of the output state. In the noisy environment, Alice and Bob implement measurements on qubits 2, 4 and 7, 9, b, respectively. The output state can be calculated as ρ ( )   The fidelity is at most 1 by definition and 0.5 is automatically achieved by outputting a completely mixed state 26 . By calculating the fidelities of the output states, we conclude that fidelity depends on the decoherence rate. In amplitude-damping and phase-damping noisy environments, we assume that = = p p p a p . The diagram of fidelity changes with the decoherence rate given in Fig. 3. We can see that the fidelities F F , A P of the output states decreases as the decoherence rate increases.

Discussions
First, we give a summary of this scheme, including the necessary operations, classical communication costs (CCCs) and efficiency. Second, we give some discussions with other schemes [18][19][20][21] . Last, the conclusions are described.
In this paper, we construct the quantum channel at first. The necessary operations have six H operations and ten CNOT operations. Quantum gate operations are critical part of quantum computation and quantum communications. Many experimental results [27][28][29][30][31] have been proposed for quantum operations implementation. For example, in 2018, Rosenblum et al. 27 realized a CNOT gate between two multiphoton qubits in two microwave cavities. They coupled two encoded qubits together through a transmon, which is driven by an RF pump to apple the gate within 190 ns. This is two orders of magnitude shorter than the decoherence time of the transmon, enabling a high-fidelity gate operation. Zajac et al. 28  than 99%, as verified by randomized benchmarking. They used the CNOT gate to generate a Bell state with 78% fidelity. According to the relevant experimental results, the implementation of the H and CNOT operations can be completed by using the existing technology. Furthermore, we can realize the construction of the eleven-qubit entangled state. In our ACBRSP scheme, no auxiliary operations are needed. The CCCs are generated when three participants send their measurement results through the classical channel. In the scheme, the measurement results can be transmitted directly to others via broadcast, since it does not contain the information about the prepared state. Thus, the CCCs are 6 bits. The efficiency 32 of the scheme can be calculated as where q s denotes the number of qubits that consist of the quantum information to be prepared, q u is the number of the qubits that is used as the quantum channel and b t is the classical bits transmitted. We give some discussions with the other ACBRSP schemes [18][19][20][21] . The results are shown in Table 1. Firstly, in refs 18,19,21 , these schemes are asymmetrical bidirectional remote preparation of an arbitrary single-and two-qubit state. The scheme in ref. 20 is asymmetrical bidirectional remote preparation of single-and four-qubit state, but in fact they only transmit the single-and two-qubit state and the receiver needs two local auxiliary qubits and auxiliary operations to recovery four-qubit state. However, in our ACBRSP scheme, we prepare a two-and three-qubit equatorial state. Secondly, we construct the quantum channel by using H operations and CNOT operations. Neither of them gave the process of the construction of quantum channel. Thirdly, in refs [19][20][21] , these schemes need the auxiliary qubits and auxiliary operations to complete the preparation task, while our scheme and the scheme in ref. 18 do not need. Fourthly, the CCCs of our scheme are 6 bits, which is less than that in refs 20,21 . Finally, the efficiency of our scheme is higher than that of other schemes.
In the future, we hope the scheme can play a facilitating role in quantum network communication. 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 [33][34][35][36] . As regards the quantum networks, the feasibility and construction have been fully verified theoretically 37,38 . 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.