Multiple phase estimation for arbitrary pure states under white noise

In any realistic quantum metrology scenarios, the ultimate precision in the estimation of parameters is limited not only by the so-called Heisenberg scaling, but also the environmental noise encountered by the underlying system. In the context of quantum estimation theory, it is of great significance to carefully evaluate the impact of a specific type of noise on the corresponding quantum Fisher information (QFI) or quantum Fisher information matrix (QFIM). Here we investigate the multiple phase estimation problem for a natural parametrization of arbitrary pure states under white noise. We obtain the explicit expression of the symmetric logarithmic derivative (SLD) and hence the analytical formula of QFIM. Moreover, the attainability of the quantum Cram\'{e}r-Rao bound (QCRB) is confirmed by the commutability of SLDs and the optimal estimators are elucidated for the experimental purpose. These findings generalize previously known partial results and highlight the role of white noise in quantum metrology.