Time operators for continuous-time and discrete-time quantum walks

TL;DR

This paper constructs explicit time operators for continuous and discrete quantum walks, analyzing their spectral properties and conditions for self-adjointness, revealing a link between winding number and spectrum discreteness.

Contribution

It introduces concrete time operators for quantum walks and characterizes their spectral and deficiency index properties, especially relating self-adjointness to topological winding numbers.

Findings

Time operators constructed for both continuous and discrete quantum walks.

Self-adjointness of the discrete-time time operator depends on the winding number.

Spectrum of the time operator is discrete when it is self-adjoint.

Abstract

We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint if the time evolution operator has a non-zero winding number. In this case, its spectrum becomes a discrete set of real numbers.