We study multifractal properties in time evolution of a single particle subject to repeated measurements. For quantum systems, we consider circuit models consisting of local unitary gates and local projective measurements. For classical systems, we consider models for estimating the trajectory of a particle evolved under local transition processes by partially measuring particle occupations. In both cases, multifractal behaviors appear in the ensemble of wave functions or probability distributions conditioned on measurement outcomes after a sufficiently long time. While the nature of particle transport (diffusive or ballistic) qualitatively affects the multifractal properties, they are even quantitatively robust to the measurement rate or specific protocols. On the other hand, multifractality is generically lost by generalized measurements allowing erroneous outcomes or by postselection of the outcomes with no particle detection. We demonstrate these properties by numerical simulations and also propose several simplified models, which allow us to analytically obtain multifractal properties in the monitored single-particle systems. Published by the American Physical Society 2024
We study multifractal properties in time evolution of a single particle subject to repeated measurements. For quantum systems, we consider circuit models consisting of local unitary gates and local projective measurements. For classical systems, we consider models for estimating the trajectory of a particle evolved under local transition processes by partially measuring particle occupations. In both cases, multifractal behaviors appear in the ensemble of wave functions or probability distributions conditioned on measurement outcomes after a sufficiently long time. While the nature of particle transport (diffusive or ballistic) qualitatively affects the multifractal properties, they are even quantitatively robust to the measurement rate or specific protocols. On the other hand, multifractality is generically lost by generalized measurements allowing erroneous outcomes or by postselection of the outcomes with no particle detection. We demonstrate these properties by numerical simulations and also propose several simplified models, which allow us to analytically obtain multifractal properties in the monitored single-particle systems. Published by the American Physical Society 2024
Repeated measurements can induce entanglement phase transitions in the dynamics of quantum systems. Interacting models, both chaotic and integrable, generically show a stable volume-law entangled phase at low measurement rates that disappears for free, Gaussian fermions. Interactions break the Gaussianity of a dynamical map in its unitary part, but non-Gaussianity can be introduced through measurements as well. By comparing the entanglement and non-Gaussianity structure of different protocols, we propose a single particle indicator of the measurement-induced phase transition, and we use it to argue in favor of the stability of the transition when non-Gaussianity is purely provided by measurements. Published by the American Physical Society 2024
This paper investigates the geometry of randomness, specifically focusing on characterizing fractal behavior using quantum simulation. The core claim is to develop a novel quantum simulation technique to unveil the underlying fractal structure of random systems, thereby providing a deeper understanding of their complex dynamics. The method aims to emulate the evolution of a random system through quantum mechanics, allowing us to directly observe and quantify the emergence of fractal patterns. The research will explore the potential of this approach to surpass current approximation methods, offering a fundamentally new pathway to analyze and quantify fractal geometry.
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. Published by the American Physical Society 2024
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024
Quantum entanglement, a phenomenon where two or more particles become inextricably linked regardless of distance, holds significant promise for understanding and potentially manipulating chaos. This paper explores the application of entanglement measurements to predict and dampen temporal chaos within complex systems exhibiting fractal geometry. We propose a novel metric for quantifying 'temporal coherence' in entangled states, linking this metric to the emergence of chaotic behavior. Our work demonstrates that calibrated entanglement measurements, when correlated with specific topological defects, can accurately forecast and suppress temporal chaos in fractal systems, offering a fundamentally new approach to modeling and control. This advancement leverages the unique properties of quantum entanglement to create a predictive tool, moving beyond traditional chaos theory's reliance on simplified models.
We investigate phase transitions in the encoding of quantum information in a quantum many-body system due to the competing effects of unitary scrambling and boundary dissipation. Specifically, we study the fate of quantum information in a one-dimensional qudit chain, subject to local unitary quantum circuit evolution in the presence of depolarizing noise at the boundary. If the qudit chain initially contains a finite amount of locally accessible quantum information, unitary evolution in the presence of boundary dissipation allows this information to remain partially protected when the dissipation is sufficiently weak, and up to timescales growing linearly in the system size L . In contrast, for strong enough dissipation, this information is completely lost to the dissipative environment. We analytically investigate this “quantum coding transition” by considering dynamics involving Haar-random, local unitary gates, and confirm our predictions in numerical simulations of Clifford quantum circuits. Scrambling the quantum information in the qudit chain with a unitary circuit of depth O ( log L ) before the onset of dissipation can perfectly protect the information until late times. The nature of the coding transition changes when the dynamics extend for times much longer than L . We further show that at weak dissipation, it is possible to code at a finite rate, i.e., a fraction of the many-body Hilbert space of the qudit chain can be used to encode quantum information. Published by the American Physical Society 2024