michael_nielsen

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.