Localization, fractality, and ergodicity in a monitored qubit

Understand

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.