Quantum Fisher information and symmetric logarithmic derivative via anti-commutators

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.