Recurrence for discrete time unitary evolutions

TL;DR

This paper investigates the recurrence properties of quantum systems with discrete time unitary evolutions, linking spectral measures to return probabilities and providing a topological interpretation of expected return times.

Contribution

It establishes the equivalence between recurrence and the spectral measure's absence of an absolutely continuous part, and interprets first return times via Schur functions and spectral theory.

Findings

Recurrence is characterized by spectral measure properties.

Expected first return time is an integer or infinite, with a topological interpretation.

First arrival amplitudes relate to Schur function coefficients.

Abstract

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.