  Symmetric logarithmic derivative (SLD) is a key quantity to obtain quantum
Fisher information (QFI) and to construct the corresponding optimal
measurements. Here we develop a method to calculate the SLD and QFI via
anti-commutators. This method is originated from the Lyapunov representation
and would be very useful for cases that the anti-commutators among the state
and its partial derivative exhibits periodic properties. As an application, we
discuss a class of states, whose squares linearly depend on the states
themselves, and give the corresponding analytical expressions of SLD and QFI. A
noisy scenario of this class of states is also considered and discussed.
Finally, we readily apply the method to the block-diagonal states and the
multi-parameter estimation problems.
