Localization, fractality, and ergodicity in a monitored qubit

TL;DR

This paper investigates the complex dynamics of a monitored qubit under repeated measurements, revealing phases of localization, delocalization, and fractality, and drawing analogies with Anderson localization to understand measurement-induced phase transitions.

Contribution

It introduces a detailed analysis of the phase diagram of a monitored qubit, identifying ergodic, nonergodic, and localized phases, and combines analytical and numerical methods for comprehensive understanding.

Findings

Identification of ergodic and nonergodic phases in monitored qubits

Discovery of a localized phase with delta-function distributions

Observation of phase transitions analogous to Anderson localization

Abstract

We study the statistical properties of a single two-level system (qubit) subject to repetitive ancilla-based measurements. This setup is a fundamental minimal model for exploring the intricate interplay between the unitary dynamics of the system and the nonunitary stochasticity introduced by quantum measurements, which is central to the phenomenon of measurement-induced phase transitions. We demonstrate that this "toy model" harbors remarkably rich dynamics, manifesting in the distribution function of the qubit's quantum states in the long-time limit. We uncover a compelling analogy with the phenomenon of Anderson localization, albeit governed by distinct underlying mechanisms. Specifically, the state distribution function of the monitored qubit, parameterized by a single angle on the Bloch sphere, exhibits diverse types of behavior familiar from the theory of Anderson transitions, spanning from complete localization to almost uniform delocalization, with fractality occurring between the two limits. By combining analytical solutions for various special cases with two complementary numerical approaches, we achieve a comprehensive understanding of the structure delineating the "phase diagram" of the model. We categorize and quantify the emergent regimes and identify two distinct phases of the monitored qubit: ergodic and nonergodic. Furthermore, we identify a genuinely localized phase within the nonergodic phase, where the state distribution functions consist of delta peaks, as opposed to the delocalized phase characterized by extended distributions. Identification of these phases and demonstration of transitions between them in a monitored qubit are our main findings.