In this post I’m going to give a clean definition of idealized quantum Brownian motion and give a few entry points into the literature surrounding its abstract formulation. A follow-up post will give an interpretation to the components in the corresponding dynamical equation, and some discussion of how the model can be generalized to take into account the ways the idealization may break down in the real world.
I needed to learn this background for a paper I am working on, and I was motivated to compile it here because the idiosyncratic results returned by Google searches, and especially this MathOverflow question (which I’ve answered), made it clear that a bird’s eye view is not easy to find. All of the material below is available in the work of other authors, but not logically developed in the way I would prefer.
Quantum Brownian motion (QBM) is a prototypical and idealized case of a quantum system , consisting of a continuous degree of freedom, that is interacting with a large multi-partite environment , in general leading to varying degrees of dissipation, dispersion, and decoherence of the system. Intuitively, the distinguishing characteristics of QBM is Markovian dynamics induced by the cumulative effect of an environment with many independent, individually weak, and (crucially) “phase-space local” components.… [continue reading]
For the upteenth time I have read a paper introducing the Wigner function essentially like this:
The Wigner-representation of a quantum state is a real-valued function on phase space definedActually, they usually use a more confusing definition. See my post on the intuitive definition of the Wigner function. a (with ) as
It’s sort of like a probability distribution because the marginals reproduce the probabilities for position and momentum measurements:
But the reason it’s not a real probability distribution is that it can be negative.
The fact that can be negative is obviously a reason you can’t think about it as a true PDF, but the marginals property is a terribly weak justification for thinking about as a “quasi-PDF”. There are all sorts of functions one could write down that would have this same property but wouldn’t encode much information about actual phase space structure, e.g., the Jigner“Jess” + “Wigner” = “Jigner”. Ha! b function , which tells as nothing whatsoever about how position relates to momentum.
Here is the real reason you should think the Wigner function is almost, but not quite, a phase-space PDF for a state :
Consider an arbitrary length scale , which determines a corresponding momentum scale and a corresponding setNot just a set of states, actually, but a Parseval tight frame.
… [continue reading]
[Follow-up post: In what sense is the Wigner function a quasiprobability distribution?]
I’ve never liked how people introduce the Wigner function (aka the Wigner quasi-probability distribution). Usually, they just write down a definition like
and say that it’s the “closest phase-space representation” of a quantum state. One immediately wonders: What’s with the weird factor of , and what the heck is ? Usually, the only justification given for the probability interpretation is that integrating over one of the variables recovers the probability distribution for the other (if it were measured):
where is just the density matrix in the momentum basis. But of course, that doesn’t really tell us why we should think of , as having anything to do with the (rough) value of conditional on a (rough) value of .
Well now I have a much better idea of what the Wigner function actually is and how to interpret it. We start by writing it down in sane variables (and suppress ):
So the first step in the interpretation is to consider the function
which appears in the integrand. This is just the (position-space) density matrix in rotated coordinates and .… [continue reading]