An underappreciated gem: probability backflow
I was recently reminded of a beautiful paper by Tony Bracken and Geoff Melloy, Probability Backflow and a New Dimensionless Quantum Number (publisher subscription required, alas). What they describe is a marvellous (and too little known) quantum phenomenon that they dub “probability backflow”.
The basic idea is simple, although counterintuitive. Imagine a quantum particle moving in free space. It’s easiest if you imagine space is one-dimensional, i.e., the particle can only move left or right on a line. The discussion also works in three dimensions, but is a bit trickier to describe.
What Bracken and Melloy show is that it’s possible to prepare the particle in a state such that both the following are true:
(1) If you measure the velocity of the particle, you’re guaranteed to find that the particle is moving rightwards along the line.
(2) Suppose you fix a co-ordinate origin on the line, i.e., a “zero” of position. They prove that if you measure position then the probability of finding the particle to the right of the origin actually decreases in time, not increases, as one might naively expect from point (1).
So far as I know, this has never been experimentally demonstrated.
Bracken and Melloy also give a beautiful explanation of probability backflow in terms of “quasi-probabilities”. It’s been known for a long time, at least since Wigner, that you can reformulate quantum mechanics as something very much like a classical theory, but one with negative probabilities. Using this reformulation, Bracken and Melloy explain the phenomenon: sure, the particle is moving right, but it’s actually negative probability that is flowing to the right of the origin, and so the probability of being to the right of the origin decreases in time.
Inspired by the power of this intuitive explanation, I once worked for a while trying to use quasi-probability formulations to find new algorithms for quantum computers. Despite my lack of success, I still think this is a promising approach, not just to algorithms, but more generally to quantum information.