Quantum teleportation plays a significant role in the quantum communication and has drawn more and more attention in recent decades due to its potential applications.^[1–6] Quantum teleportation, first proposed by Bennett et al. in 1993,^[7] could transmit quantum information from a source particle to a destination particle by quantum entanglement. Shared entanglement can be achieved by generating an EPR pair and distributing this pair to the source and destination in advance.^[8] The first experiment was realized in 1997, and a quantum teleportation distance of 143 kilometers has been achieved by Ma et al.^[9]

Long-distance delivery of quantum information is an essential challenge in the future quantum network. Considering the quantum entanglement state can be generated and distributed between smaller segments, quantum repeaters are used as a router to share and distribute quantum entanglement.^[10] In recent decades, lots of innovative approaches have been proposed in the quantum teleportation domain.^[11–15] In recent years, some significant research works have been done in the domain of multi-hop quantum teleportation. Wang et al.^[16] analyzed the multi-hop teleportation based on arbitrary Bell pairs using EPR-pair bridging. The possibility of multi-hop teleportation based on partially entangled state has a relationship with the number of the nodes N. If N is odd, the possibility of successful transmission , where K is the hops of the transmission.^[17] If N is even, the destination node will add another teleportation to itself.^[18] Dur et al.^[19] studied the use of N pairs Werner states for quantum communication over long distances, but they focused on purification. Su et al.^[20] proposed a quantum teleportation scheme using double W states and its successful possibility is after one hop teleportation. Shi et al.^[21] proposed a quantum wireless multi-hop network and the information is transmitted from the source to the destination using Werner states as the quantum channel directly.

In this paper, we proposed a multi-hop quantum transmission scheme using the combination of W state and EPR pairs and proved its feasibility. The quantum information is transmitted hop by hop via intermediate nodes through the quantum channel established by W state and EPR pairs. We assume that the sender intends to teleport an unknown two-level quantum state: |ψ⟩ = α |00⟩ + β |11⟩ to the receiver. Firstly, we deduce the possibility of successful transmission after one hop teleportation. The recovered qubits are used as the teleported states for the next-hop teleportation. Based on the results of one-hop and two-hop teleportation, we make an assumption about multi-hop possibility and use mathematical induction to prove this assumption. Finally, we obtain a higher possibility of after n-hop teleportation compared to the schemes only using double W states.

The rest of the paper is organized as follows. In Sections 2 and 3, the details of the multi-hop teleportation are discussed and proved, including one-hop, two-hop, and multi-hop teleportation. The conclusions are presented in Section 4. Operations of the Pauli operator used in this paper are introduced in Appendix A.

The fidelity of the quantum entanglement decreases as the distance increases in the long-distance quantum transmission. It is practical to transmit quantum states from source to destination in a multi-hop way with more than one intermediate state. In the following two sections, a multi-hop transmission method based on W state and EPR pairs is presented and its validity is verified.

There are two nodes in Fig. 1. Alice is the source node and Bob is the destination node. Alice holds particles 1, 2, 3, and 6. Bob holds particles 4, 5, 7, and 8. Particle 8 is an auxiliary particle. The quantum state to be transmitted is |ψ₁₂⟩ = α |00⟩ ₁₂ + β |11⟩ ₁₂, where α and β satisfy α² + β² = 1. Particles 6 and 7 share an EPR pair represented as . Particles 3–5 are in a W state represented as: . The W state and EPR pair constitute the quantum channel.

Fig. 1.

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	Fig. 1. One-hop teleportation.

The process of quantum state transmission is described as follows.

According to this process, the entire quantum circuit is given in Fig. 2. The solid line represents quantum data and the wavy line represents classical measurement results.

Fig. 2.

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	Fig. 2. One-hop quantum circuit.

The entire system can be represented as

Considering the following equation:

After the Bell measurement on particles 1 and 3, the possibility of getting is

The possibility of getting is

Assuming the measurement result is , ignoring the coefficient, then the system state becomes

The possibility of getting , , , and is

Assuming the measurement result is , then the system state is

At last, after the measurement, if the |0⟩₄ is obtained with the possibility of , the system state would turn into: α|10⟩ ₅₇ + β|01⟩ ₅₇ and the possibility is

Every destination state can be turned into

Table 1.

Table 1.

Table 1. Results of one-hop teleportation for |ψ⟩ 12 ⊗ |W⟩ 345 ⊗ |ψ⟩ 67. . Bell measurement result ofparticles 1,3 and 2,6 Result ofparticle 4 Destinationstate Possibility |0⟩4 α|10⟩57 + β|01⟩57 |0⟩4 α|10⟩57 − β|01⟩57 |0⟩4 α|10⟩57 + β|01⟩57 |0⟩4 α|10⟩57 + β|01⟩57 |0⟩4 α|00⟩57 + β|11⟩57 |0⟩4 α|00⟩57 − β|11⟩57 |0⟩4 α|01⟩57 + β|10⟩57 |0⟩4 α|01⟩57 − β|10⟩57

Table 1. Results of one-hop teleportation for |ψ⟩ 12 ⊗ |W⟩ 345 ⊗ |ψ⟩ 67. .

We assume that quantum information at the source node is teleported through a multi-hop quantum channel based on W state and EPR pairs and we deduce the expression of each type of the results at the destination node and its possibility. We also introduce an auxiliary particle and give the corresponding operation of the Pauli operator to recover the message. The possibility of each type is and we can recover the transmitted message with the possibility of after n-hop teleportation. There are still some issues worthy of study. In this paper, quantum channels are established by maximally entangled states. In practice, the maximally entangled state usually degrades and changes to a partially entangled state. A scheme using partially entangled states is worth studying. Meanwhile, a more efficient approach by introducing more auxiliary particles is also worth studying.