Over the past few decades, research into monitored quantum systems has progressed rapidly. Recent experimental progress in simulating and detecting quantum dynamics has considerably deepened our understanding of open quantum systems. Unlike standard open systems, where information is lost to the environment, monitored quantum systems allow observers to retain access to measurement outcomes during non-unitary dynamics. These stochastic outcomes can be tracked continuously, giving rise to quantum trajectories that provide detailed insight into open quantum dynamics. Importantly, many quantum trajectories exhibit universal or “typical” behaviors for certain physical quantities.

To shed light on this active research frontier, a recent review published in Progress of Theoretical and Experimental Physics presents an introduction to monitored quantum systems and quantum trajectories, while surveying recent developments concerning their spectral structure, typical behavior, and phases.

Beginning with the basic formalism of quantum measurement theory, the review explains how stochastic measurement outcomes generate individual quantum trajectories and how averaging over these trajectories yields the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation. The article further discusses how to predict steady states and relaxation dynamics governed by the GKSL equation.

A major highlight of the review is its focus on typical properties of quantum trajectories, such as ergodicity and purification. The review provides simple examples to explain the physical intuition for these concepts. Furthermore, it introduces and highlights typicality for trajectory-based quantities, such as Lyapunov exponents, and explains how they can be used to characterize novel measurement-induced phase transitions in many-body systems. The review also proposes several directions for future research.

These advances in understanding quantum trajectories and open quantum systems may contribute to more precise control of quantum systems. Such insights may aid in the creation and stabilization of quantum states relevant for emerging quantum technologies like quantum computing and quantum sensing.

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