  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.
