Fisher information under decoherence in Bloch representation

The dynamics of two variants of quantum Fisher information under decoherence is investigated from a geometrical point of view. We first derive the explicit formulas of these two quantities for a single qubit in terms of the Bloch vector. Moreover, we obtain analytical results for them under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of both sets of information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further express the two information quantities in terms of the Bloch vector for a qudit by expanding the density matrix and Hermitian operators in a common set of generators of the Lie algebra su (d). By calculating the dynamical quantum Fisher information, we find that the collisional dephasing significantly diminishes the precision of the phase parameter with the Ramsey interferometry.