Teleportation Fidelity of Binary Tree Quantum Repeater Networks

Abstract

Binary tree network, being a subclass of Cayley tree network, is a significant topological structure used for information transfer in a hierarchical sense. In this article, we consider four types of binary tree repeater networks (directed and undirected, asymmetric and symmetric) and obtain the analytical expressions of the average of the maximum teleportation fidelities for each of these binary tree networks. We contribute a methodology for the analytical calculation of pathlengths in all considered graph types. Based on these, we have used simple Werner state-based models and are able to identify the parameter ranges for which these networks can show quantum advantage. We also explore the role of maximally entangled states in the network to enhance the quantum advantage. We provide a detailed examination of the large-scale behavior of these networks, obtaining the limiting value of the average maximum teleportation fidelity as the number of nodes, $N$, approaches infinity, same as fractal tree. Our findings reveal that the directed symmetric binary tree represents the most advantageous topology for quantum teleportation within this context. From the context of quantum repeater networks, this work makes a significant advancement in the process of identifying resourceful tree networks for distributed quantum teleportation i.e. teleportation between all possible sources and targets.

Figures (14)

• Figure 1: Entanglement Swapping: (a) Alice and Charlie are not connected, but entangled pairs are shared between Alice-Bob and Bob-Charlie. (b) Bob's measurement creates entanglement between Alice and Charlie. (c) Sequential protocol to establish entanglement between $N+2$ nodes using $N$ swaps.
• Figure 2: Top Panel: We have (a) DABT and (b) DSBT with 7 and 15 nodes, respectively. Bottom Panel: We have (c) UABT and (d) USBT with 7 and 15 nodes, respectively. The depth of each binary tree is $d$ = 3.
• Figure 3: The fractal tree from an undirected symmetric binary tree. From left to right, the depths of the binary trees are 2, 4, 6, 8, and 10, respectively.
• Figure 4: Toy Example: In this UABT network with 7 nodes, the first connection is established between 1 and 6 (shown using the red line). After that, the connection is established between 1 and 5 (orange line) and finally between 1 and 7 (green line).
• Figure 5: The plot of $F^{\text{tel}}_{\text{avg}}$ vs for different values of $p$ in case of all four types of binary trees (a) for $N = 15$ (In case of DABT and UABT, $d = 7$ and for DSBT and USBT, $d = 3$), (b) $N=127$. (In case of DABT and UABT, $d = 63$ and for DSBT and USBT, $d = 6$). Here, the blue, green, red and orange lines denote DABT, DSBT, UABT and USBT, respectively.