Optimal quantum Fisher information for infinite dimensional systems I'm new to the topic of quantum estimation and quantum Fisher information. The papers I've seen on the topic mostly deals with Hamiltonians with a finite number of energy levels, like a collection of spins. A common formula for the optimal quantum fisher information is given in terms of the difference of the maximal and minimal eigenvalues of an operator related to the Hamiltonian.
My question is, what about systems whose Hilbert space is infinite-dimensional?, such systems does not have a maximal eigenvalue, obviously. It is there a general formula for the optimal fisher information in this case?.
 A: Generally speaking, the QFI is defined as
$$
\mathcal{F}_q = Tr[\rho_\lambda L_\lambda^2]
$$
where $L_\lambda$ is the Symmetric Logarithm derivative implicitly defined as
$$
2\partial_\lambda \rho_\lambda = \rho_\lambda L_\lambda + L_\lambda \rho_\lambda
$$
The QFI fixes a lower bound on the estimation of the parameter $\lambda$, i.e. the variance of any estimator $\hat{\lambda}$ must satisfy
$$
Var(\hat{\lambda}) \geq \mathcal{F}_q^{-1}
$$
The case you depicted is a very special case in which

*

*only pure states are considered

*the parameter is encoded by a unitary $\mathcal{U} = e^{-i\gamma \mathcal{H}}$, thus $\vert \psi_\gamma \rangle = e^{-i\gamma \mathcal{H}}\vert\psi_0\rangle$
From these assumption one can maximise the $\mathcal{F}_q = \langle \psi_\gamma \vert L_\gamma^2 \vert \psi_\gamma \rangle$ on the set of initial state $\vert \psi_0\rangle$ and eventually finds out what you stated.
Concerning your question on system with infinite-dimensional Hilbert space (CV hence after), things are more difficult, as one can expect. As far as I know, a general expression for CV systems of the QFI is not known.
However, if one restricts to gaussian states, then a recipe for the calculations of the QFI can be found here.
As you see, you may try to optimise your state by considering the optimising on the initial covariant matrix and first moments, but this problem strongly depends on what kind of parameters are you dealing with. Maybe, for some class of problems analytical calculations are possible.
