Absence of Ballistic Transport in Quantum Walks with Asymptotically Reflecting Sites

TL;DR

This paper establishes conditions under which one-dimensional quantum walks exhibit zero velocity, indicating the absence of ballistic transport, applicable to both deterministic and random cases, and valid in the CMV framework.

Contribution

It provides general sufficient criteria for zero velocity in quantum walks based on site structure and coin parameters, extending to random and CMV cases.

Findings

Derived deterministic criteria for zero velocity based on coin entries.

Established that certain site structures lead to absence of ballistic transport.

Extended results to random quantum walks and the CMV setting.

Abstract

We prove general sufficient conditions for zero velocity in position dependent one-dimensional quantum walks, and hence for the absence of ballistic transport. Our starting point is a general a priori upper bound on the velocity, formulated in terms of sparse bi-infinite sequences of sites, their gap structure, and the corresponding local coin parameters. This estimate yields several deterministic criteria for zero velocity that depend only on the behavior of suitable coin entries along selected subsequences and are independent of the values of the coins elsewhere. We also discuss the random case as an application of the general approach. All of our results remain valid in the CMV setting.