General intuition:
The classical fisher information (CFI) is a measure of how quickly a probability distribution changes with respect to some parameter. While the quantum fisher information(QFI) is how quickly a quantum state (represented by a density matrix) changes with respect to some parameter. To define such a measure, one needs to a define a distance on the manifold of probability distributions or quantum states (Projective Hilbert Space). For a probability distribution such a metric can be fixed by a set of subtle mathematical assumptions but in general the direct expression for the fisher information is more illuminating:
$
F_c(\theta)=\sum_x P(x,\theta)[\frac{1}{P(x,\theta)}\frac{d}{d\theta}P(x,\theta)]^2
$
This is the average of the percent change of each $P(x,\theta)$ for a small amount of change in $\theta$. The percent change, $\frac{1}{P(x,\theta)}\frac{d}{d\theta}P(x,\theta)$ is squared because otherwise the average would be 0.
The QFI can be constructed from this expression. First, notice that to obtain a probability distribution of a quantum state, you must measure in a particular basis, say $R$. Then by optimizing the CFI over that observable R we obtain the quantum fisher information.
$
F_q(\theta)=max_R F_c(\rho(\theta),R)
$
By thinking about the Brunes Metric and specifying how $\rho$ depends on $\theta$ one can relate this expression to one easy for calculations. If we define $\rho(\theta)=e^{iQ\theta}\rho e^{-iQ\theta}$, it's possible to derive:
$
F_q(Q,\theta=0)= 2\sum_{l,l'}\frac{(p_{l}-p_{l'})^{2}}{p_{l}+p_{l'}}\left|\left<l\right|Q\left|l'\right>\right|^2
$
where $\left| l\right>$ are the eigenvectors of the density matrix and $p_{l}$ are the eigenvalues.
I don't know this proof so you'll have to look for articles if you interested. But it's not necessary for interpretation. Simply from the optimization of the CFI, you can see it is some quantification of how quickly a quantum state changes with respect to the parameter $\theta$.
Now the article you linked makes really only a few substitutions to connect the QFI to the linear response of a thermal state $\rho=e^{-\beta H}/Z$ to a perturbation $Q$ driven at various frequencies. Thinking about the QFI as quantifying how quickly a state changes with respect to some operation Hamiltonian $Q$ gives some intuition about why it might be related to response functions.
Entanglement
Relating the QFI to entanglement is done in just a few steps.
1st) one uses the fact that the QFI of a pure state is 4 times the variance of $Q$: $F_q(Q,\theta=0)=4(\left<Q^2\right>-\left<Q\right>^2)$ (Easily proved from the above expression).
2nd) one uses the fact that the quantum fisher information is convex in the space of density matricies. That is if I have two pure states $\rho_a=\left|a\right>\left<a\right|$ and $\rho_b=\left|b\right>\left<b\right|$ the convex sum of them has a lower QFI:
$
F_q(p\rho_a+(1-p)\rho_b)<pF_Q(\rho_a)+(1-p)F_q(\rho_b)
$
for $0<p<1$ and any $Q$. Thus for a mixed state, the QFI will be limited by the QFI of the eigenstate with maximum variance in Q.
The final step is to bound the variance of an unentangled state for a particular $Q$. Suppose I break my system into $N$ parts for which I want to quantify the entanglement between. For example, I could take the parts as the $N$ qubits in a quantum computer or the $N$ particles in a system. I will now choose $Q$ to be a sum of observable of the individual parts: $Q=\sum_{i=1}^N Q_i$. I will also assume these operators to be bounded and have norm $|Q_i|=1$(magnitude of the maximum eigenvalue of $Q_i$ is 1). For a untangled pure state,the statistics of the parts are independent so the maximum variance is $N$. Due to convexity, the maximum fisher information for a unentangled mixed state is then 4N. Therefore a fisher information $F_q>4N$ is a witness of entanglement.
There is also has been work to argue that $\max_Q F_q(Q)/4N$ is a approximation of the number of parts which must be mutually entangled. I can't find the work that does this but they do it by working with generalizations of the GHZ state
Is there a prototypical model Hamiltonian/system that has nontrivial information encoded in its QFI?
The QFI has general applicability to all quantum states (such as those produced by a sequence of Unitaries in a quantum computer) rather then equilibrium states of physical systems. Therefore I would claim that the prototypical states for discussing quantum fisher information are the GHZ state (the characteristic entangled state) and coherent states(the characteristic unentangled state). Since they are pure states, it's straight forward to calculate there variance and show that the GHZ state has maximum QFI $F_q=4N^2$ for the "maximum" observable, $Q$ and the coherent state has QFI $F_q(Q)=4N$. Again assuming $Q=\sum_iQ_i$ and $|Q_i|=1$.
I believe the review suggested above("Introduction to quantum Fisher information" (14 Aug 2010), by Petz and Ghinea) has a more careful/precise analysis of these states.
