# Quantum Fisher information for Gaussian states without eigenproblem?

Given a Gaussian-preserving interaction (including a unitary operation and losses) for a Gaussian input, I want to know if there is an "easy" way to compute the Quantum Fisher information without having to find the solutions to the eigenproblem related to the measured (output) state.

To be more specific, given a probe $$\rho$$, I have a Gaussian-preserving unitary interaction $$U_\lambda$$ that parametrizes my probe. Considering also that there are losses (let's call the lost channel $$L$$), the measured state is $$\rho_\lambda := Tr_L (U_\lambda \rho U_\lambda^\dagger).$$

If the eigensolutions for the matrix $$\rho_\lambda$$ are $$\lbrace \rho_m, |\rho_m\rangle \rbrace_m$$, then the Quantum Fisher Information is given by

$$H = 2\sum_{m,n}\frac{|\langle \rho_m|\partial_\lambda \rho_\lambda|\rho_n\rangle|^2}{\rho_m + \rho_n},$$ where the sum runs for all $$m,n$$ such that $$\rho_m + \rho_n \neq 0$$.

My question is whether there is a way to avoid the eigenproblem in the case of Gaussian input + Gaussian preserving channel.

• I found the answer in case anyone is interested. Safranek has some papers on this. And also the book by Serafini on Continuous Quantum Variables – MCV Jan 24 at 16:26