Spectral Properties of Quantum Walks on Rooted Binary Trees
Spectral Properties of Quantum Walks on Rooted Binary Trees
复制标题
DOI：
10.1007/s10955-014-0950-x
复制
发表时间：
2013-07
影响因子：
1.6
通讯作者：
A. Joye;L. Marin
中科院分区：
文献类型：
--
作者：
A. Joye;L. Marin
我们定义了无限维Hilbert空间上由么正算子给出的无限有根二叉树上的造量子游动，并研究了它们的谱性质。对于循环酉币矩阵，我们导出了与循环向量的谱测度有关的Carathéodory函数的一个方程。这使得我们可以证明，对于所有循环酉币矩阵，量子游动的谱没有奇异的连续分量。此外，对于正交循环矩阵的硬币矩阵，我们证明了除了四个硬币矩阵的谱是纯点外，量子游动的谱是绝对连续的。
We define coined quantum walks on the infinite rooted binary tree given by unitary operatorson an associated infinite dimensional Hilbert space, depending on a unitary coin matrix, and study their spectral properties. For circulant unitary coin matrices, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for. This allows us to show that for all circulant unitary coin matrices, the spectrum of the quantum walk has no singular continuous component. Furthermore, for coin matriceswhich are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum ofis pure point.