Preprocessing quantum states for noisy measurements in quantum metrology

Quantum Fisher information (QFI) characterizes the amount of information a quantum state carries about an unknown parameter, assuming arbitrary quantum measurements can be applied on the quantum state. However, in practice, quantum measurements are usually noisy and cannot attain the QFI of a given quantum state. Here we study the metrological protocol where quantum states can be preprocessed using quantum controls before noisy measurements. We formulate the problem of identifying the optimal quantum channels to be applied on a quantum state that maximize the classical Fisher information (FI) of the noisy measurement statistics as a biconvex optimization by introducing the concept of error observables. Based on this formulation, for pure states, we prove unitary channels are optimal and also derive analytical solutions to the optimal controls in a few practically relevant cases. For classically mixed states (i.e., states of which the unknown parameter is encoded in the eigenvalues) with commuting measurement operators, we prove that coarse graining channels are optimal and provide a counter example where unitary controls are not optimal. For general quantum states and measurements, we provide useful upper and lower bounds on the FI optimized over preprocessing controls. Finally, we consider quantum states in a multi-partite system with local noisy measurements acting independently on each subsystem and prove that in the asymptotic limit, the QFI is attainable using global optimal controls for a generic class of quantum states.