Isotropic quantum walks on lattices and the Weyl equation

TL;DR

This paper classifies isotropic quantum walks on 1D, 2D, and 3D lattices, showing that in 3D only Weyl quantum walks exist, and derives the Weyl equation from informational principles using isotropy.

Contribution

It provides a comprehensive classification of isotropic quantum walks on lattices, simplifying previous derivations and removing unnecessary assumptions.

Findings

In 3D, only Weyl quantum walks are isotropic.

The Weyl equation is derived from informational principles.

The classification is shorter and more general than previous work.

Abstract

We present a thorough classification of the isotropic quantum walks on lattices of dimension $d=1,2,3$ for cell dimension $s=2$. For $d=3$ there exist two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.