Just dug up one of my favourite papers, one I haven't looked at in years! Richard Feynman's paper trying to make sense of negative probabilities: http://cds.cern.ch/record/154856/files/pre-27827.pdf?version=1

This might sound nuts. Actually, it has some really nice applications. For instance, there's a _really_ beautiful application known as "probability backflow".

Turns out that you can find quantum states of a particle (in 1 dimension) so that with certainty the particle is moving to the right. _But_ - and this is the crazy bit - in fact the probability the particle is found to the right of the origin actually _decreases_ with time.

This sounds impossible. But if what's happening is that it's _negative probability_ which is all flowing to the right, then it makes sense. This idea was developed in this lovely paper by Bracken and Melloy https://people.smp.uq.edu.au/TonyBracken/backflow1.pdf

If you feel like you didn't get that the first time, you are not alone. You really need to read it half a dozen times for it to even parse. But it's a real feature of the world!

Often wondered if this might be a good way for developing good intuitions for new quantum algorithms - the idea is to explore many possibilities, and then to use negative probabilities to "unexplore" fruitless directions. But I never made it work usefully.

Let me try to unpack that description of probability backflow just a little more, so it makes more sense. You have a particle in one dimension - think of it as moving on a line, left to right. It has the following properties:

(1) If you measure the velocity, you're absolutely guaranteed to find that it is positive (i.e., moving to the right); and (2) nonetheless, the probability the particle is to the right of the origin actually _decreases_ over time.

That sounds just straight up impossible - if something is guaranteed to be moving to the right, it can't be less likely that it's to the right of the origin over time!

Well, there's a description of quantum particles based on what's called a quasiprobability function that actually makes it work. It provides a kind of "probability" p(x, v) that the particle has position x and velocity v.

Turns out - I wish I had a movie to show you, it makes it much easier to understand (hi @3blue1brown ) - that what's going on is that small amounts of _negative_ quasiprobability are flowing to the right, & that's why the probability of being to the right of the origin decreases

Now, how to think about the negative probabilities themselves? Well, quantum mechanics tells you that you can't ever observe position and velocity simultaneously. So there's no need to find a direct interpretation.

Still, that feels like a copout to me - I think there probably is a really good, clear interpretation of what it means. I'm not sure what that is, unfortunately!