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

TL;DR

This review addresses how monitoring quantum systems reshapes dynamics through quantum trajectories, linking the spectral properties of outcome-averaged CPTP maps to the typical, trajectory-dependent behaviors such as ergodicity and purification. It develops a coherent framework from projective and indirect measurements to CP-instruments and GKSL dynamics, highlighting how Lyapunov exponents and spectral gaps govern convergence and purification, and how these notions signal measurement-induced phase transitions in many-body settings. The article also surveys numerical unraveling methods, nonlinear trajectory observables, and emergent dynamical topologies, emphasizing the distinct physics visible at the level of individual trajectories versus ensemble-averaged dynamics. Overall, it provides a rigorous-physicist bridge between spectral theory, stochastic quantum dynamics, and non-equilibrium phases induced by measurements, with practical implications for simulating open quantum systems and diagnosing dynamical phase transitions.

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 the 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.

Figures (20)

• Figure 1: (a) Indirect measurement for the case with a pure state. We let the state $\hat{\rho}=\ket{\psi}\bra{\psi}$ of the system and the state $\hat{\sigma}_\mathrm{M}=\ket{0}\bra{0}$ of the meter interact via $\hat{U}_\mathrm{int}$ and measure the meter's state. If the outcome is $b$, the post-measurement state for the system becomes $\hat{\rho}_b'=\ket{\psi_b'}\bra{\psi_b'}$. (b) Repeated measurement: we repeat the above process (a) and obtain the quantum trajectory $\hat{\rho}_{\bm{b};n}=\ket{\psi_{\bm{b};n}}\bra{\psi_{\bm{b};n}}$ depending on the measurement outcomes.
• Figure 2: Schematic illustration of a quantum trajectory. Starting from $\ket{\psi(0)}$, the state undergoes a non-Hermitian continuous time evolution governed by $\hat{H}_\mathrm{eff}$, unless a quantum jump described by the jump operator $\hat{L}_b$ occurs, which suddenly changes the state. Here, the jump times are denoted as $\tau_1$ and $\tau_2$.
• Figure 3: An example of a quantum trajectory for a single atom system with spontaneous emissions. (a) If we consider the occupation of the excited state, given as $\frac{\hat{\sigma}^z+1}{2}$, it is described by the continuous time evolution and quantum jumps, which reset the state into the ground state. (b) The total number of jumps $N(t)$ until time $t$. It increases by one at times $\tau_1$, $\tau_2$, and $\tau_3$.
• Figure 4: Sampling of a random variable $X$, which takes $X_b\;(0\leq b\leq B)$, with probability $p_b$. Considering the cumulative distribution $Q_b$ and choosing $R$ randomly from $R\in [0,1]$, we can sample $X_b$ from the condition $Q_{b-1}<R\leq Q_b$.
• Figure 5: (a) When a CPTP map $\mathcal{E}$ is irreducible, it has a nondegenerate peripheral spectrum $z \in \{ e^{2\pi i k/m} \}_{k=0}^{m-1}$ ($m=6$ in this example). (b) When a CPTP map $\mathcal{E}$ is primitive, $z=1$ is the only eigenvalue with unit modulus.

Theorems & Definitions (19)

• Proposition 1: Irreducibility
• Theorem 2: Irreducibility in Kraus representation
• Theorem 3: Irreducibility from spectral properties
• Theorem 4: Peripheral spectrum of irreducible CPTP maps
• Theorem 5: Irreducibility from steady state
• Proposition 6: Primitivity
• Theorem 7: Primitivity in Kraus representation
• Proposition 8: Irreducibility of Markovian CPTP maps
• Theorem 9: Wolf wolf2012quantum, Yoshida yoshida2024uniqueness, Zhang-Barthel zhang2024criteria
• Theorem 10: Wolf wolf2012quantum