Arrival and First Passage Times for Quantum Random Walks

A number of authors have developed quantum analogues of the random walk models used to model evidence accumulation. The original analysis is due to Busemeyer, Wang and Townsend (2006) and an interesting recent application was presented by Kvam et al (2015).

In classical evidence accumulation models the knowledge state evolves until it reached a threshold, at which point a decision is taken. The time at which this happens can be found by computing the first passage time at the threshold for the random walk.

An as yet unstudied problem is how to extract similar predictions from a quantum random walk model. The situation is complicated because the analogue of the first passage time in quantum theory does not exist. This is part of a general problem often known as the ‘problem of time’ in quantum theory, which concerns the difficulty one has in defining ‘time-like’ quantities, such as arrival time, dwell time, in quantum theory.

One can get a handle on why the problem is complicated by noting that ‘crossing a threshold’ amounts to making two statements about the system, a) It is at the threshold, b) It is moving across it. In other words, one is trying to say something about the position and the momentum of the system simultaneously, and this is forbidden in quantum theory.

One amusing consequence of this is the following: Suppose you have a quantum state in 1D (i.e. on a line) that consists entirely of positive momentum. It is not the case that the probability of being in x>0 is a non-decreasing function of time! This is known as the quantum ‘backflow’ effect, and can be used to prove that an ideal absorbing boundary is impossible in quantum theory (ideal here means it absorbs all wave function inside it while leaving the wave function outside unchanged.)

An introduction to these issues can be found here