Randomly repeated measurements on quantum systems: Correlations and topological invariants of the quantum evolution

TL;DR

This paper investigates how random measurements in quantum systems lead to quantized mean return times, linking these to topological invariants like the Berry phase, and revealing a linear scaling with Hilbert space dimension.

Contribution

It introduces a topological explanation for the quantization of mean return times in monitored quantum evolutions, connecting measurement statistics to Berry phases.

Findings

Mean number of measurements until first detection equals Hilbert space dimension

Mean return time scales linearly with Hilbert space dimension

Quantization of return time explained via Berry phase

Abstract

Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return time is equal to the average time step between successive measurements times the mean number of measurements. Thus, the mean first detected return time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.