Quantum Communication in Acyclic Directed Networks

Abstract

We propose conditions for establishing the teleportation protocol between any two nodes in a quantum network. The conditions include (i) a network connection must exist between any two nodes, in other words, the network transfer matrix must be non-singluar, and (ii) the unitary acting on the quantum state after measurement during teleportation must be the inverse of the unitary operator that corresponds to the classical outcome from the composition of encoding functions through the intermediate nodes. These conditions are verified in a butterfly network and over a general k-pair network, which we term as the Gk network with k sender/receiver pairs. Our analysis is based on line graphs, proved under certain conditions. The non-singularity of the transfer matrices of these networks is essential for the establishment of a connection between senderreceiver pairs towards quantum teleportation. A fidelity analysis over the Gk network is also performed to assess the fidelity loss when prior entanglement is absent. It is well understood that prior entanglement is indispensable for the perfect transmission of quantum states. These results are useful for paving the way towards entanglement-assisted quantum communication networks. Further, we studied the teleportation over networks with non-maximally entangled states by using the quantum state bridging circuit. The teleportation is probabilistic and hence, a method for recovering state without information loss was proposed. The method is also applied in the case of a butterfly network, and the probability of successful recovery of the transferred state is derived