Local Probability Conservation in Discrete Time Quantum Walks

TL;DR

This paper demonstrates that probability is locally conserved in discrete time quantum walks on arbitrary directed graphs, by defining probability currents that respect the locality structure and yield correct distributions.

Contribution

It introduces a framework for defining local probability currents in discrete quantum walks on arbitrary directed graphs, ensuring local conservation of probability.

Findings

Probability currents can be defined respecting the graph's locality.

The framework applies to any unitary evolution respecting locality.

Correct final probability distributions are obtained through these currents.

Abstract

We show that probability is locally conserved in discrete time quantum walks, corresponding to a particle evolving in discrete space and time. In particular, for a spatial structure represented by an arbitrary directed graph, and any unitary evolution of a particle which respects that locality structure, we can define probability currents which also respect the locality structure and which yield the correct final probability distribution.