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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 $\rho$?

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