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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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    $\begingroup$ One can write $\rho$ in terms of $W$ with the aid of a Weyl transformation. Have you tried that? $\endgroup$ Aug 5 at 6:52
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    $\begingroup$ I don't think the "QFI with respect to a quadrature measurement" makes sense. The QFI is a property of a parametrised family of quantum states, and is defined as the optimum over all possible measurement bases. Perhaps what you are really after is just the regular ("classical") Fisher information for a quadrature measurement? In this case you should easily be able to get what you need from the Wigner function, whose marginals yield a genuine probability distribution for the quadratures $x,p$ (e.g. $P(x) = \int W(x,p) dp$). $\endgroup$ Aug 5 at 7:00
  • $\begingroup$ In case you are unfamiliar with the Wikipedia formula, $\rho= \frac{2}{(2\pi \hbar)^{3/2}}\iint \!\!\!\iint\!\! dq\, dp\, d\tilde{x} \, d\tilde{p} \ e^{ \frac{i}{\hbar} (\tilde {x} \tilde {p} -2(\tilde{p}-p)(\tilde{x}-q))}~ W(q,p) ~ |\tilde{x}\rangle\langle \tilde{p}|$. If you provided a simplest example/ challenge in your question, you might get a better answer, instead of nebulous generalities. $\endgroup$ Aug 5 at 19:12
  • $\begingroup$ @MarkMitchison So this might be a core misunderstanding of mine about the use of QFI. Using the QFI definition that requires diagonalizing the density matrix (which is the first definition on the Wikipedia page) it defines the QFI with respect to a state and an observable, and in my case I'm doing a homodyne measurement of $\hat X$. Is it sufficient then to, as you suggest, simply look at the marginal and find the classical fisher information? Would that give me the same thing as finding $\rho$, diagonalizing, and then taking my hermitian observable to be $\hat X$? $\endgroup$ Aug 5 at 20:23
  • $\begingroup$ Thanks for the replies. I'm aware of the way in which one obtains $\rho$ from $W$. I suppose I failed to mention that another reason I don't want to go that route is that the two main ways of calculating the QFI involve diagonalizing $\rho$, or exponentiating $\rho$ (arxiv.org/pdf/1801.00945.pdf) both of which I expect to be difficult in my case. $\endgroup$ Aug 5 at 20:27

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