arXiv:1708.00087v6 [quant-ph] 24 Feb 2018 Investigating quantum wireless multihop teleportation under decoherence * Marlon David Gonz´ alez Ram´ ırez, 1 Christian Nwachioma, 2, 1 Adenike Grace Adepoju, 1 and Babatunde James Falaye 3, † 1 CIDETEC, Instituto Polit´ ecnico Nacional, UPALM, CDMX 07700, Mexico 2 Department of Physics, COMSATS Institute of Information Technology, Park Road, Islamabad 45550, Pakistan. 3 Applied Theoretical Physics Division, Department of Physics, Federal University Lafia, P. M. B. 146, Lafia, Nigeria (Dated: November 11, 2021) This research work scrutinizes quantum routing protocol with multihop teleportation for wireless mesh backbone networks, in amplitude and phase damping channels. After analyzing the quantum multihop protocol, we select a four-qubit cluster state as the quantum channel for the protocol. The quantum channel linking the intermediate nodes has been established via entanglement swapping based on four-qubit cluster state. Also, we established the classical and the quantum route in a distributed manner. We show that from the source node to the destination node, quantum information can be teleported hop-by-hop through an amplitude damping channel. We show that the quantum teleportation could be successful if the sender node performs Bell state measurements (BSM), and the receiver introduces auxiliary particles, applies positive operative value measure and then utilizes corresponding unitary transformation to recover the transmitted state. We scrutinize the success probability of transferring the quantum state through a noisy channel. We found that optimum probability would be attained if decoherence rate of amplitude damping channel (ξa) is zero or the number of hops (N ) is above 75. Our numerical results evince susceptibility of success probability to ξa and N . It has been shown that as the decoherence increases, the fidelity exponentially decays until it vanishes. This decay is as a consequence of information loss from the system to the surrounding. However, the fidelity can be enhanced by considering fewer hops. Keywords: Multihop teleportation; Amplitude damping channel; Phase damping channel; Four- qubit cluster state; Fidelity. PACS numbers: 03.67.Hk, 03.65.Ud, 42.50.-p, 03.67.-a, 03.65.Yz. I. INTRODUCTION In the recent years, there has been incessant avidity in studying multi-user quantum communication because it offers the opportunity to construct quantum networks. With quantum networks, quantum information between physically separate quantum systems can be transmitted. In fact, it forms a salient component of quantum comput- ing and quantum cryptography systems. It has been dis- cerned that transmission via quantum teleportation and, directly from one node to another are two methods to transmit an unknown quantum state between two nodes [1]. In this paper, the later, i.e., the node-to-node trans- mission will be considered. Since quantum systems un- avoidably interact with the environment, node-to-node transmission easily debase the quantum states. Con- sequently, it would be reasonable to take the influence of noise into consideration while investigating quantum wireless multihop teleportation. Recently, a scheme for faithful quantum communica- tion in quantum wireless multihop networks, by perform- ing quantum teleportation between two distant nodes * https://arxiv.org/abs/1708.00087 † Electronic address: babatunde.falaye@gmail.com which do not initially share entanglement with each other, was proposed by Wang et al. [2]. It has been found in Ref. [4] that wireless quantum networks can be established between nodes of different hops sharing two qubit states. Xiong et al. [3] proposed a quantum com- munication network model where a mesh backbone net- work structure was introduced. The entanglement source deployment problem has been scrutinized by Zou et al. [5] in a quantum multihop network, which has a notable impact on quantum connectivity. Some other outstanding reports can be found in Refs. ([6–9] and references therein). Although several rele- vant results have been obtained along this direction, most contributions have been based on 2- and 3-qubit Greenberger-Horne-Zeilinger (GHZ) state as the quan- tum channel in a closed quantum system. However, quantum systems cannot but interact with the environ- ment. These unavoidable interactions make quantum systems to lose some of their properties. In this paper, we examine a quantum routing protocol with multihop tele- portation for wireless mesh backbone networks, based on four-qubit cluster state, in an amplitude damping chan- nel, which can be induced experimentally. The cluster state [10], which is a type of highly entan- gled state of multiple qubits, is generated in lattices of qubits with Ising type interactions. On the basis of single qubit operation, the cluster state serves as the initial re-