Dynamics of Unitary Operators

TL;DR

This paper investigates the spreading behavior of unitary operators on Hilbert spaces, establishing lower bounds for transport exponents linked to spectral measures, with applications to quantum walks and CMV matrices.

Contribution

It extends the analysis of dynamical spreading from Schrödinger operators to unitary operators, providing new bounds and methods for models like quantum walks.

Findings

Lower bounds for transport exponents in unitary dynamics.

Application of subordinacy theory to CMV matrices.

Explicit bounds for Fibonacci quantum walk.

Abstract

We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the time-averaged spreading on a power-law scale, in terms of dimensional properties of the spectral measure associated with the unitary operator and the initial state. These results are the unitary analog of results established in recent years for the dynamics of the Schr\"odinger equation, which is a continuum-time dynamical system associated with a self-adjoint operator. We discuss how these general results may be studied by means of subordinacy theory in cases where the unitary operator is given by a CMV matrix. An example of particular interest in which this scenario arises is given by a time-homogeneous quantum walk on the integers. For the particular case of the time-homogeneous Fibonacci quantum walk, we illustrate how these components work together and produce explicit lower bounds for the transport exponents associated with this model.