In theory the Wigner function $W(q,p)$ contains all the same information as the density matrix $\rho$ associated with it, so it's definitely possible to write the QFI $\mathcal F_Q$ in terms of the Wigner function. But all the expressions I've found in the literature for the QFI involves either diagonalizing $\rho$ or exponentiating $\rho$, both of which I expect to be difficult in my case.
The motivation for me is that the classical Fisher information can be written as an integral over a probability distribution, and it seems like the QFI should be able to be written in some simple way as an integral over the Wigner function (or some quasiprobability distritution), but maybe that's a naive assumption.
Edit: More specifically, I am considering a density operator $\rho(t)$ that describes a bosonic mode, and I would like to calculate the QFI with respect to the parameter $\theta$ and the family of states $\rho_\theta=e^{i\theta\hat X}\rho_0e^{-i\theta\hat X}$. I’m wondering if there’s a way to do that without finding $\rho$ from $W(q,p)$.
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1 Answers
So I think I've figured out the answer and what I was misunderstanding. I was already planning on doing an $P$ basis measurement of the state, with respect to translations generated by $X$, but I now understand that in such a case it isn't sensical to talk about the QFI, since the QFI is the maximum over all measurements (and I was already set on a measurement). What I was really imagining was just the classical Fisher information.
As for whether or not there is a way to obtain the QFI directly from the Wigner function without obtaining $\rho$, the answer seems to be that (outside of special cases) one needs to obtain $\rho$ and then calculate the QFI from there. One of those special cases is if you happen to know the optimal POVM for your family of states, in which case the QFI is just the CFI of the probability distribution associated with that POVM.
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