Optimal operation for fidelity estimation of shared entangled states with arbitrary noise and no prior information

Fidelity estimation for entangled states is an essential building block of quantum networks. Practical quantum networks often encounter heterogeneous and correlated noise. Thus, the design of low-error fidelity estimation protocol in the presence of arbitrary noise is an interesting avenue of research. Because a measured state can no longer be used by quantum applications, this work considers a scenario in which the nodes randomly sample part of the entangled qubit pairs for measurement to estimate the fidelity of the unsampled pairs conditioned on the measurement outcome. The study reveals that by evaluating the fidelity conditioned on the measurement outcome, a post-selection of the qubit states is introduced. When a proper measurement operation is selected, this post-selection effect neutralizes the detrimental effect of arbitrary noise. Based on this finding, the proposed measurement operation for fidelity estimation achieves the minimum estimation error in scenarios with arbitrary noise and no prior information. Analysis shows that by using the proposed operation, the estimation efficiency with arbitrary noise is of the same order as that with i.i.d. noise, demostrating the effectivness of the proposed approach in containing the detrimental effect of arbitrary noise.