D'Ariano G M, Mosco N, Perinotti P, Tosini A
QUIT Group, Dipartimento di Fisica, Via Bassi 6, 27100 Pavia, Italy.
QUIT Group, Dipartimento di Fisica, Via Bassi 6, 27100 Pavia, Italy
Philos Trans A Math Phys Eng Sci. 2017 Nov 13;375(2106). doi: 10.1098/rsta.2016.0394.
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group [Formula: see text], which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of [Formula: see text] that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.This article is part of the themed issue 'Second quantum revolution: foundational questions'.
我们考虑3 + 1维的外尔量子行走，这是一种离散时间行走，描述了一个具有两个内禀自由度的粒子在群[公式：见正文]的凯莱图上移动，在适当的条件下，它根据外尔方程演化。外尔量子行走最近被推导为群[公式：见正文]的凯莱图上唯一的均匀且各向同性的酉演化。本文通过传播子的解析表达式，即在位置表象中给出了量子行走演化的一般解，传播子即从图的一个节点到另一个节点在有限步数内的跃迁振幅。行走的量子特性体现在连接给定节点的图上路径的干涉中。该解基于图上允许路径的二进制编码以及行走转移矩阵的半群结构。本文是主题为“第二次量子革命：基础问题”特刊的一部分。
