  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.
