Error minimization for fidelity estimation of GHZ states with arbitrary noise

Fidelity estimation is an essential building block for the quality control of entanglement distribution networks. This work considers a scenario in which multiple nodes share noisy GHZ states. Because measurements collapse quantum states, the nodes randomly sample a subset of noisy GHZ states for measurement and then estimate the average fidelity of the unsampled states conditioned on the measurement outcome. By constructing a fidelity-preserving diagonalization operation, analyzing the Bloch representation of GHZ states, and maximizing the Fisher information, the proposed estimation protocol achieves the lowest mean squared estimation error in a difficult scenario with arbitrary noise and no prior information. Moreover, this protocol is implementation friendly as it only performs local Pauli operators according to a predefined sequence. Numerical studies show that compared to existing fidelity estimation protocols, the proposed protocol reduces the estimation error in both scenarios with i.i.d. noise and correlated noise.