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arXiv:2512.19922 · Progress of Theoretical and Experimental Physics

An introduction to monitored quantum systems and quantum trajectories: spectrum, typicality, and phases

Ryusuke Hamazaki, Ken Mochizuki, Hisanori Oshima, Yohei Fuji

cond-mat.stat-mechmath-phquant-phSubmitted Dec 22, 2025• Announced Dec 24, 2025• 2 citations• Revised Mar 21, 2026 (v2)
v1 · Dec 22, 2025v2 · Mar 21, 2026
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Abstract

Thanks to recent experimental advances in simulating and detecting quantum dynamics with high precision and controllability, our understanding of the physics of monitored quantum systems has considerably deepened over the past decades. In this article, we provide an introductory theoretical review on the basic formalisms governing open quantum dynamics under measurement, along with recent developments in their spectral and typical aspects. After reviewing quantum measurement theory, we introduce the concept of quantum trajectories, which are the conditional dynamics of monitored states shaped by a set of measurement outcomes. We then discuss the spectral properties of the dynamical map describing the evolution averaged over measurement outcomes. As has recently been recognized, these spectral features are intimately connected to whether quantum trajectories exhibit typical behaviors, such as ergodicity and purification. Moreover, we introduce Lyapunov exponents of typical quantum trajectories and discuss how these quantities serve as indicators of measurement-induced phase transitions in monitored quantum many-body systems.

Journal reference
Progress of Theoretical and Experimental Physics, ptag055 (2026)
DOI
10.1093/ptep/ptag055
Comments
Accepted version; 162 pages, 20 figures

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During recent years the interest to dynamics of quantum systems has grown considerably. Quantum many body systems out of equilibrium often manifest behavior, different from the one predicted by standard statistical mechanics and thermodynamics in equilibrium. Since the dynamics of a many-body quantum system typically involve many excited eigenstates, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last decade such new subjects like quantum quenches, thermalization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of investigation in modern quantum physics. One of the most interesting themes in the study of dynamics of quantum many-body systems out of equilibrium is connected with the recently proposed important concept of dynamical quantum phase transitions. During the last few years a great progress has been achieved in studying of those singularities in the time dependence of characteristics of quantum mechanical systems, in particular, in understanding how the quantum critical points of equilibrium thermodynamics affect their dynamical properties. Dynamical quantum phase transitions reveal universality, scaling, connection to the topology, and many other interesting features. Here we review the recent achievements of this quickly developing part of low-temperature quantum physics. The study of dynamical quantum phase transitions is especially important in context of their connection to the problem of the modern theory of quantum information, where namely non-equilibrium dynamics of many-body quantum system plays the major role.
Abstract We explore the Lyapunov spectrum and entanglement entropy in quantum trajectories evolved by quantum measurements and spatially homogeneous unitary gates. In models with temporally random and Floquet unitary gates, we find that the Lyapunov exponents typically converge to values independent of measurement outcomes and that spectral transitions of the Lyapunov spectrum and entanglement transitions of ground states occur at the same measurement thresholds. Our results indicate that (i) randomness due to quantum measurements is enough, or equivalently, randomness of unitary gates is not essential, for the typical convergence of the Lyapunov spectrum and that (ii) the correspondence between the spectral transition and entanglement transition is ubiquitous in a wide range of systems. We also discuss why the typical convergence occurs, in light of the uniqueness and positive definiteness of stationary states in the completely-positive trace-preserving (CPTP) dynamics averaged over random measurement outcomes. Our discussions indicate that properties of the averaged CPTP dynamics play a key role in characterizing measurement-induced transitions in quantum trajectories.
Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook to major open questions as well as future directions of research.