Recurrence Time for Finite Quantum Systems

TL;DR

This paper investigates the recurrence time in finite quantum systems, establishing bounds using number theory and approximation techniques to understand when all states return close to their initial configuration.

Contribution

It introduces a formal definition of recurrence time for finite quantum systems and derives tighter bounds using Dirichlet's approximation theorem.

Findings

Derived bounds on recurrence time for quantum systems

Linked recurrence time to rational approximation of real numbers

Provided mathematical results to tighten recurrence time bounds

Abstract

We study the time it takes for all states of a finite quantum system to return simultaneously to their original configuration. In particular, we define the recurrence time for a quantum system to be the time at which all time-evolved states are close to their initial configuration, and at least one state has deviated significantly during this interval. Considering finite-dimensional quantum systems evolving unitarily, we find bounds on this notion of recurrence time, for continuous time and discrete time, by using Dirichlet's approximation theorem. We show how the problem of finding a bound on recurrence time can be related to approximating the difference of real numbers by rationals. We present a mathematical result on the latter, which we then use to obtain tighter bounds on recurrence time.