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Now I'll tell you a bit about what it actually says!
Remember the story so far: we've got a physical system that's in a state of maximum entropy. I didn't emphasize this yet, but that happens whenever our system is in thermodynamic equilibrium. An example would be a box of gas inside a piston. Suppose you choose any number for the energy of the gas and any number for its volume. Then there's a unique state of the gas that maximizes its entropy, given the constraint that on average, its energy and volume have the values you've chosen. And this describes what the gas will be like in equilibrium!
Remember, by 'state' I mean mixed state: it's a probabilistic description. And I say the energy and volume have chosen values on average because there will be random fluctuations. Indeed, if you look carefully at the head of the piston, you'll see it quivering: the volume of the gas only equals the volume you've specified on average. Same for the energy.
More generally: imagine picking any list of numbers, and finding the maximum entropy state where some chosen observables have these numbers as their average values. Then there will be fluctuations in the values of these observables — thermal fluctuations, but also possibly quantum fluctuations. So, you'll get a probability distribution on the space of possible values of your chosen observables. You should visualize this probability distribution as a little fuzzy cloud centered at the average value!
To a first approximation, this cloud will be shaped like a little ellipsoid. And if you can pick the average value of your observables to be whatever you'll like, you'll get lots of little ellipsoids this way, one centered at each point. And the cool idea is to imagine the space of possible values of your observables as having a weirdly warped geometry, such that relative to this geometry, these ellipsoids are actually spheres.
This weirdly warped geometry is an example of an 'information geometry': a geometry that's defined using the concept of information. This shouldn't be surprising: after all, we're talking about maximum entropy, and entropy is related to information. But I want to gradually make this idea more precise. Bring on the math!
We've got a bunch of observables
This state
lying in some open subset of
By the way, I should really call this Gibbs state
Now at each point
If we take its real part, we get a symmetric matrix:
It's also nonnegative — that's easy to see, since the variance of a probability distribution can't be negative. When we're lucky this matrix will be positive definite. When we're even luckier, it will depend smoothly on
So far this is all review of last time. Sorry: I seem to have reached the age where I can't say anything interesting without warming up for about 15 minutes first. It's like when my mom tells me about an exciting event that happened to her: she starts by saying "Well, I woke up, and it was cloudy out..."
But now I want to give you an explicit formula for the metric
Crooks does the classical case — so let's do the quantum case, okay? Last time I claimed that in the quantum case, our maximum-entropy state is the Gibbs state
where
(To be honest: last time I wrote the indices on the conjugate variables
Also last time I claimed that it's tremendously fun and enlightening to take the derivative of the logarithm of
But now let's take the derivative of the logarithm of
we get
Next, let's differentiate both sides with respect to
Hey! Now we've got a formula for the 'fluctuation' of the observable
This is incredibly cool! I should have learned this formula decades ago, but somehow I just bumped into it now. I knew of course that
But I never had the brains to think about
Now we get our cool formula for
But now that we know
we get the formula we were looking for:
Beautiful, eh? And of course the expected value of any observable
so we can also write the covariance matrix like this:
Lo and behold! This formula makes sense whenever
Indeed, whenever we have any smooth function from a manifold to the space of density matrices for some Hilbert space, we can define
The classical analogue is the somewhat more well-known 'Fisher information metric'. When we go from quantum to classical, operators become functions and traces become integrals. There's nothing complex anymore, so taking the real part becomes unnecessary. So the Fisher information metric looks like this:
Here I'm assuming we've got a smooth function
Crooks says more: he describes an experiment that would let you measure the length of a path with respect to the Fisher information metric — at least in the case where the state
There's a lot more to say about this, and also about another question: What use is the Fisher information metric in the general case where the states
But it's dinnertime, so I'll stop here.
You can read a discussion of this article on Azimuth, and make your own comments or ask questions there!
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