Parity-enhanced quantum optimal measurements

In quantum metrology, measurement and estimation schemes are vital for achieving higher precision, along with initial state preparation. This article presents the compound measurement of parity and particle number, which is optimal for a broad range of states named equator states (ESs). ES encompasses most pure input states used in current studies and, more significantly, a wide range of mixed states. Moreover, the ES can be prepared directly using non-demolition parity measurement. We thus propose an improved quantum phase estimation protocol applicable to arbitrary input states, ensuring precision consistently surpassing that of the standard protocol. The proposed scheme is also demonstrated using a nonlinear interferometer, with the realization of the non-demolition parity measurement in atomic condensates.