Characteristics of multi-hop noisy quantum entanglement channel and optimal relay protocol

Abstract

The optimal relay path calculation and selection are important factors to affect the performance of quantum communication network. Current researches seldom consider the quantum path selection in real noisy environments. One of the difficult problems is how to analyze the influence of the noise on the quantum communication in multi-hop channels. This paper aims to solve the path selection problem of the quantum teleportation network in noisy environments. The process of entanglement swapping in the phase damping channel is first studied with an example of two-hop quantum channel, whose damping factors are p1 and p2. The entanglement states |φ> 12+ and |φ> 34+ are distributed separately in each hop. After the entanglement swapping, the density matrix of the entanglement state of photon 1 and photon 4 is obtained by performing a partial trace over the environment. Then, the Bures fidelity of this entanglement is calculated. After that, we define the path equivalent damping factor to describe the characteristic of the two-hop noisy quantum relay path. With an equivalent calculation method, the results above can be generalized to multi-hop channel. The path equivalent damping factor of the multi-hop amplitude damping channel is also obtained. According to these results, we propose an optimal relay protocol for the quantum teleportation network with the criterion of path equivalent damping factor, which means that a relay path with the minimum path equivalent damping factor can obtain the highest teleportation fidelity. The types and parameters of the messages used in the protocol are given. The processes of the relay protocol are described specifically, including neighbor finding, quantum link noise measurement, and quantum link status transmission. An improved Dijkstra algorithm is used in the optimal path calculation. Furthermore, because the entanglement resources maintained in the quantum nodes are limited and may be exhausted in superior quantum links, we propose a resource reservation method to avoid the failure of the relay path setup. Theoretical analysis and simulation show that our method can obtain a lower average equivalent damping factor and higher teleportation fidelity. It can also be seen that increasing the number of the entanglement resources will raise the performance of the quantum network, however, it brings higher cost and complexity. Therefore, the entanglement resources maintained in the quantum nodes must be configured reasonably according to the network scale, the cost, the time delay and the need of the users.