# Fisher information of parametric channel

Suppose $$\Phi_\theta$$ is a quantum channel whose action can be written for any state $$\rho\in \mathcal S(\mathcal H_S)$$ in the Stinespring representation as $$\Phi_\theta(\rho)= \text{Tr}_E(U_\theta (\rho_S\otimes \tau_E) U_\theta^\dagger)$$ for some unitary $$U_\theta$$ living on the total space $$S\otimes E$$. Suppose also that, if $$a = (a_1, b_1,...,a_n, b_n)^t$$ is the vector of the combined modes describing the system $$S$$ (so that the initial state of the system $$\rho_S$$ can be written as some function $$\rho_S=f(a, a^\dagger)$$), such unitary is described by its action on them $$U_\theta a^\dagger U_\theta^\dagger=u_\theta a^\dagger$$ for some matrix $$u_\theta$$.

The quantum Fisher information of a state $$\rho_\theta$$ is generally defined as $$H(\theta) = \text{Tr}(\partial_\theta \rho_\theta L_\theta)$$, where $$L_\theta$$ is the SLD operator. More explicitly, there exists the following formula $$H(\theta) = 2\int_0^{+\infty} dt \ e^{-\rho_\theta t}\partial_\theta \rho_\theta e^{-\rho_\theta t}$$ which only involves $$\rho_\theta$$ and its derivatives. Assuming knowledge of $$u_\theta$$, is there a simple way to express the QFI for a channel such as $$\Phi_\theta$$, in terms of any initial state $$\rho$$?