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Just dug up one of my favourite papers, one I haven& #39;t looked at in years! Richard Feynman& #39;s paper trying to make sense of negative probabilities: http://cds.cern.ch/record/154856/files/pre-27827.pdf?version=1">https://cds.cern.ch/record/15...
This might sound nuts. Actually, it has some really nice applications. For instance, there& #39;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& #39;s happening is that it& #39;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">https://people.smp.uq.edu.au/TonyBrack...
If you feel like you didn& #39;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& #39;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& #39;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& #39;t be less likely that it& #39;s to the right of the origin over time!
Well, there& #39;s a description of quantum particles based on what& #39;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& #39;s going on is that small amounts of _negative_ quasiprobability are flowing to the right, & that& #39;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& #39;t ever observe position and velocity simultaneously. So there& #39;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& #39;m not sure what that is, unfortunately!
