Localization, fractality, and ergodicity in a monitored qubit

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.

49 pages, 23 figures. Comments welcome

References in corpus (17)

• Anderson Transitions
• Critical properties of the measurement-induced transition in random quantum circuits
• Postselection-free entanglement dynamics via spacetime duality
• Measurement-induced entanglement and teleportation on a noisy quantum processor
• Entanglement Transition in the Projective Transverse Field Ising Model
• Theory of free fermions under random projective measurements
• Nonlinear sigma models for monitored dynamics of free fermions
• Measurement-induced phase transition for free fermions above one dimension
• Weak values of electron spin in a double quantum dot
• Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?
• Purification Timescales in Monitored Fermions
• Measurement-altered Ising quantum criticality
• Tomography of many-body weak values: Mach-Zehnder interferometry
• Return probability and scaling exponents in the critical random matrix ensemble
• Levy flights and multifractality in quantum critical diffusion and in classical random walks on fractals
• Evolution of many-body systems under ancilla quantum measurements
• Engineering unsteerable quantum states with active feedback

Cited by in corpus (12)

• Measurement-induced transitions for interacting fermions
• Topological transitions in quantum jump dynamics: Hidden exceptional points
• Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field
• Generalizing measurement-induced phase transitions to information exchange symmetry breaking
• Ancilla quantum measurements on interacting chains: Sensitivity of entanglement dynamics to the type and concentration of detectors
• Universal Stochastic Equations of Monitored Quantum Dynamics
• Entanglement dynamics of monitored noninteracting fermions on graphics processing units
• Mesoscopic Fluctuations and Multifractality at and across Measurement-Induced Phase Transition
• Enhanced localization in the prethermal regime of continuously measured many-body localized systems
• Error threshold in active steering protocols for few-qubit systems
• Motion from Measurement: The Role of Symmetry of Quantum Measurements
• Generation of Volume-Law Entanglement by Local-Measurement-Only Quantum Dynamics