Precision-guaranteed quantum metrology

Quantum metrology is a general term for methods to precisely estimate the value of an unknown parameter by actively using quantum resources. In particular, some classes of entangled states can be used to significantly suppress the estimation error. Here we derive a formula for rigorously evaluating an upper bound for the estimation error in a general setting of quantum metrology with arbitrary finite data sets. Unlike in the standard approach, where lower bounds for the error are evaluated in an ideal setting with almost infinite data, our method rigorously guarantees the estimation precision in realistic settings with finite data. We also prove that our upper bound shows the Heisenberg limit scaling whenever the linearized uncertainty, which is a popular benchmark in the standard approach, shows it. As an example, we apply our result to a Ramsey interferometer, and numerically show that the upper bound can exhibit the quantum enhancement of precision for finite data.