Understanding the connection between Quantum Fisher Information and Correlation Functions

Question

This is a port of a question I've asked on physics.se, but never got a response.

This question is regarding an article in Nature Physics discussing Quantum Fisher Information (QFI):

http://www.nature.com/nphys/journal/v12/n8/full/nphys3700.html

Frankly I have not been following Quantum Information for a number of years now, and have never heard of even classical Fisher information.

What I was surprised to see was that there is a connection between the QFI and an energy integral over a response function (e.g. Optical conductivity). I normally interpret response functions as correlations between operator expectation values (e.g. current-current correlations, spin-spin, etc.), and have not considered the possible connections to entanglement.

From the text, it seems that the Quantum Fisher Information is given by the total energy integral of the response function. What puzzles me is how this could possibly be useful, for example, the total integral of the optical conductivity will give some type of QFI, but because of the optical sum rule this integral always gives the number of charge carriers, making it a useless measure of any entanglement.

So I have the following questions:

1. What is the motivation for defining (Q)FI the way it is?
2. How do you interpret QFI, especially with regards to measuring entanglement?
3. What kinds of dynamical correlation functions actually give useful measures of entanglement/QFI? The optical conductivity for example seems like it would be totally useless for example.