In quantum communication and quantum computation, quantum entanglement is a unique resource of the implementation. So far, some schemes have been proposed experimentally and theoretically to solve the problem of multi-mode entanglement works, such as multi-mode squeezing of parametrically amplified multi-wave mixing in atomic ensemble or atomic-like medium.^[1–4] Quantum communication technology is considered as an important technology in the field of quantum information, such as quantum teleportation (QT),^[5–12] remote state preparation (RSP),^[13–26] quantum key distribution (QKD),^[27–29] and so on. In RSP, the sender can transmit a known state to the receiver and it has less classical communication cost than QT. In order to improve the security in RSP, a variety of schemes of controlled RSP (CRSP),^[30–37] joint RSP (JRSP),^[38–48] controlled bidirectional RSP (CBRSP),^[49–53] and controlled JRSP (CJRSP)^[54–56] are presented.

In recent years, many protocols have been proposed for preparing equatorial states. In 2011, Chen et al.^[57] proposed a probabilistic joint remote preparation of a two-particle high-dimensional equatorial state. In 2015, Choudhury et al.^[58] presented a joint RSP scheme for preparing two-qubit equatorial states. At the same year, a scheme for joint remote preparation of multi-qubit equatorial states was proposed by Li et al.^[59] A year later, Wei et al.^[60] proposed a deterministic RSP scheme to remotely prepare arbitrary multi-qubit equatorial states. Recently, Sun et al.^[53] realized an asymmetric bidirectional transmission of two- and four-qubit equatorial states synchronously.

Inspired by some RSP schemes in Refs. [53,58–60], we present a scheme for remotely preparing arbitrary multi-qubit equatorial states from the sender Alice to the receiver Bob, in which n two-qubit maximally entangled states are used as the quantum channel. The characteristics of our scheme is that we find a simple form of measurement basis to realize the RSP.

The rest of this paper is organized as follows. In Section 2, an efficient scheme for remotely preparing n-qubit equatorial states is presented via n two-qubit maximally entangled states. Moreover, we demonstrate some concrete examples of preparing equatorial states to make our scheme clear and feasible. Finally, a brief discussion and a concluding summary are drawn in Section 3.

In order to interpret our protocol better, now we begin to realize the remote preparation of arbitrary n-qubit equatorial states. Suppose there are two participants named Alice and Bob, the former is the sender and the latter is the receiver, who are situated in separated sites. Presume that Alice wants to help Bob prepare arbitrary n-qubit equatorial states. Generally, it has the form as given below

It can be obtained from the permutation and combination calculation formula that when Alice wants to help Bob prepare arbitrary n-qubit equatorial states, the possibility of σ_z appearing in the measurement bases can be presented as

In order to show the clarity of our protocol, meanwhile, we discuss the remote preparation of three-qubit and four-qubit equatorial states respectively.

2.1. Examples of remote preparation of three-qubit equatorial state

Suppose the sender Alice would like to help the receiver Bob prepare the following three-qubit equatorial state

The quantum channel which between Alice and Bob is three two-qubit entangled states

Now, Alice measures the particles A₁, A₂, and A₃ under the basis

in which the bases are given by

With the mutual orthogonal measurement bases Eq. (8), we can rewrite the entangled state as

After Alice makes the three-particle projective measurement on her particles A₁, A₂, and A₃ under the basis

Without loss of generality, suppose Alice’s measurement result is |Γ₃⟩A₁A₂A₃, the state of particles B₁, B₂, B₃ will collapse into

From Eq. (10), Bob executes appropriate unitary operations I_B1 ⊗ σ_B2z ⊗ I_B3 on his qubits B₁, B₂, and B₃ after receiving the result from Alice. The state which Alice wants to prepare can always be obtained. Here, Alice’s measurement outcomes, Bob’s collapse states, and the corresponding unitary operations are listed in Table 1.

Table 1.

Table 1.

Table 1. The measurement outcomes of Alice, Bob’s collapse states and the unitary operators of Bob. . Alice’s measurements Bob’s collapse states Bob’s unitary operations |Γ1⟩A1A2A3 IB1 ⊗ IB2 ⊗ IB3 |Γ2⟩A1A2A3 σB1z ⊗ IB2 ⊗ IB3 |Γ3⟩A1A2A3 IB1 ⊗ σB2z ⊗ IB3 |Γ4⟩A1A2A3 IB1 ⊗ IB2 ⊗ σB3z |Γ5⟩A1A2A3 σB1z ⊗ σB2z ⊗ IB3 |Γ6⟩A1A2A3 σB1z ⊗ IB2 ⊗ σB3z |Γ7⟩A1A2A3 IB1 ⊗ σB2z ⊗ σB3z |Γ8⟩A1A2A3 σB1z ⊗ σB2z ⊗ σB3z

Table 1. The measurement outcomes of Alice, Bob’s collapse states and the unitary operators of Bob. .

2.2. Examples of remote preparation of four-qubit equatorial state

Suppose that Alice wishes to prepare a four-qubit equatorial state for Bob, which can be written as

The quantum channel can be made up of four two-qubit entangled states, which are presented as

In order to help the receiver Bob prepare the equatorial state remotely, the sender Alice needs to perform the four-qubit projective measurements on her particles A₁, A₂, A₃, and A₄ under the bases |Γ_j⟩ (j = 1,2,3,…, 16) as follows:

Thus, the quantum channel can be rewritten as

After Alice’s measurements, she sends the results to the intend receiver Bob via classical channel. If Alice’s measurement result is |Γ₁₆⟩_A1A2A3A4, the state of Bob will collapse into

Based on receiving the classical information from Alice, Bob performs appropriate unitary operations σ_B1z ⊗ σ_B2z ⊗ σ_B3z ⊗ σ_B4z to convert the collapsed state into the target state. Other cases are elucidated in Table 2.

Table 2.

Table 2.

Table 2. The measurement outcomes of Alice, Bob’s collapse states and the unitary operators of Bob. . Alice’s measurements Bob’s collapse states Bob’s unitary operations |Γ1⟩A1A2A3A4 |φ1⟩B1B2B3B4 IB1 ⊗ IB2 ⊗ IB3 ⊗ IB4 |Γ2⟩A1A2A3A4 |φ2⟩B1B2B3B4 σB1z ⊗ IB2 ⊗ IB3 ⊗ IB4 |Γ3⟩A1A2A3A4 |φ3⟩B1B2B3B4 IB1 ⊗ σB2z ⊗ IB3 ⊗ IB4 |Γ4⟩A1A2A3A4 |φ4⟩B1B2B3B4 IB1 ⊗ IB2 ⊗ σB3z ⊗ IB4 |Γ5⟩A1A2A3A4 |φ5⟩B1B2B3B4 IB1 ⊗ IB2 ⊗ IB3 ⊗ σB4z |Γ6⟩A1A2A3A4 |φ6⟩B1B2B3B4 σB1z ⊗ σB2z ⊗ IB3 ⊗ IB4 |Γ7⟩A1A2A3A4 |φ7⟩B1B2B3B4 σB1z ⊗ IB2 ⊗ σB3z ⊗ IB4 |Γ8⟩A1A2A3A4 |φ8⟩B1B2B3B4 σB1z ⊗ IB2 ⊗ IB3 ⊗ σB4z |Γ9⟩A1A2A3A4 |φ9⟩B1B2B3B4 IB1 ⊗ σB2z ⊗ σB3z ⊗ IB4 |Γ10⟩A1A2A3A4 |φ10⟩B1B2B3B4 IB1 ⊗ σB2z ⊗ IB3 ⊗ σB4z |Γ11⟩A1A2A3A4 |φ11⟩B1B2B3B4 IB1 ⊗ IB2 ⊗ σB3z ⊗ σB4z |Γ12⟩A1A2A3A4 |φ12⟩B1B2B3B4 σB1z ⊗ σB2z ⊗ σB3z ⊗ IB4 |Γ13⟩A1A2A3A4 |φ13⟩B1B2B3B4 σB1z ⊗ σB2z ⊗ IB3 ⊗ σB4z |Γ14⟩A1A2A3A4 |φ14⟩B1B2B3B4 σB1z ⊗ IB2 ⊗ σB3z ⊗ σB4z |Γ15⟩A1A2A3A4 |φ15⟩B1B2B3B4 IB1 ⊗ σB2z ⊗ σB3z ⊗ σB4z |Γ16⟩A1A2A3A4 |φ16⟩B1B2B3B4 σB1z ⊗ σB2z ⊗ σB3z ⊗ σB4z

Table 2. The measurement outcomes of Alice, Bob’s collapse states and the unitary operators of Bob. .

In this scheme, we give a new way by utilizing n two-qubit maximally entangled states as the entangled quantum channel to prepare arbitrary n-qubit equatorial states, the probability of success can be reached 100%. Compared with the other schemes,^[53,57–60] our scheme makes remote preparation of arbitrary equatorial states more simple and convenient for simplifying the measurement basis and unitary transformations in our protocol. Additionally, our scheme has no coefficient constraint and auxiliary qubits. So the success probabilities in our scheme are independent of the coefficients of the entangled channel. Furthermore, we analysis the possibility of σ_z appearing in the measurement bases. It will be helpful for the following study of equatorial states. In addition, our scheme is demonstrated by theoretical knowledge and concrete instances. The protocol also has greatly improved the efficiency, because of needing fewer appropriate unitary transformations to get the desired state. Thus, we wish our scheme will be realized by experiment in future.