Error minimization for fidelity estimation of GHZ states with arbitrary noise

Fidelity estimation is a crucial component for the quality control of entanglement distribution networks. This work studies a scenario in which multiple nodes share noisy Greenberger-Horne-Zeilinger (GHZ) states. Due to the collapsing nature of quantum measurements, 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 developing a fidelity-preserving diagonalization operation, analyzing the Bloch representation of GHZ states, and maximizing the Fisher information, the proposed estimation protocol achieves the minimum mean squared estimation error in a challenging scenario characterized by arbitrary noise and the absence of prior information. Additionally, this protocol is implementation-friendly as it only uses local Pauli operators according to a predefined sequence. Numerical studies demonstrate that, compared to existing fidelity estimation protocols, the proposed protocol reduces estimation errors in both scenarios involving independent and identically distributed (i.i.d.) noise and correlated noise.