Convex Optimization Approaches to Optimal Teleportation Fidelity in Linear Three-Party Networks

Abstract

We study the maximum achievable quantum teleportation fidelity between two distant parties, Alice and Charlie, where each of them share a bipartite quantum state only with a common intermediary, Bob, and all parties are allowed to perform {\it Local Operations and Classical Communication} (LOCC). As the structure of LOCC is complicated, we relax the set of free operations to separable (SEP) operations and formulate a convex optimization problem that provides upper bounds on the LOCC achievable fidelity value. We observe that the complexity of such optimization problem reduces significantly if we restrict ourselves to a subclass of SEP operations, where the Kraus operators of either Alice or Charlie are proportional to unitary operators, leading to a simplified convex optimization that matches the general LOCC limit for certain two-qubit states. Through explicit examples, we show that protocols initiated by Bob by performing measurements in a maximally entangled basis are not necessarily optimal, and alternative strategies can outperform them. Finally, we extend our analysis to linear networks and demonstrate that different LOCC strategies can achieve the same optimal fidelity while consuming different amounts of entanglement content.