Monitored Quantum Dynamics

• Monitored quantum dynamics is a framework combining unitary evolution with sequential measurements to reveal individual quantum trajectories beyond ensemble averages.
• It employs Kraus operators and large deviation theory to identify dynamical phase transitions, glassy intermittency, and topologically protected modes in many-body systems.
• Experimental realizations on Rydberg arrays, trapped ions, and superconducting qubits validate theoretical predictions and drive advances in quantum simulation and information theory.

Monitored quantum dynamics concerns the evolution of quantum many-body systems subjected to sequences of measurements—either continuous or discrete—interleaved with unitary evolution. Distinct from classical open-system decoherence, monitored quantum systems retain explicit records of measurement outcomes, enabling the study of individual quantum trajectories rather than mere average density matrices. This selective monitoring exposes a host of nonequilibrium phenomena, including phase coexistence, glassy intermittency, measurement-induced phase transitions, nontrivial topology in trajectory space, and rich information-theoretic structure, all now at the frontier of theoretical and experimental quantum physics.

1. Dynamical Protocols and Mathematical Framework

A general discrete-time monitored quantum protocol alternates unitary evolution (possibly with kinetic constraints or random gates) with local or global projective measurements followed by outcome readout and, optionally, resetting of ancillary degrees of freedom. For a system $S$ and an environment (ancilla) $A$, the total Hilbert space is $H_{tot} = H_S \otimes H_A$. Each time step involves: (i) preparing $A$, (ii) applying a joint unitary $U$ on $S \otimes A$, (iii) measuring $A$ in a fixed basis yielding a bit string $\vec{k}$, (iv) resetting $A$.

The conditional evolution of the system state is governed by Kraus operators $K_{\vec{k}} = \langle \vec{k}_A | U | 0_A \rangle$: $$|\psi_{t+1}\rangle \propto K_{\vec{k}(t+1)} \cdots K_{\vec{k}(1)} |\psi_0\rangle,$$ with trajectory probability $\pi(\vec{k}) = \| K_{\vec{k}} |\psi\rangle \|^2$. The average quantum channel is $$\mathcal E[\rho] = \sum_{\vec{k}} K_{\vec{k}}\rho K_{\vec{k}}^\dagger$$.

Measurement-induced dynamics can be implemented in numerous architectures, including Rydberg atom arrays, trapped ions, and superconducting qubits with mid-circuit readout capabilities (Cech et al., 2024).

2. Typical and Atypical Behavior: Stationarity, Trajectory Heterogeneity, and Dynamical Phases

The ensemble-averaged map $\mathcal{E}$ typically possesses a trivial infinite-temperature stationary state, for example, $\rho_\infty = 1/2^L$ for $L$ qubits with local ancilla monitoring. However, individual quantum trajectories exhibit persistent heterogeneity and collective structure not visible in ensemble averages.

Dynamical order parameters are constructed from the space-time record of measurement outcomes, such as the activity,

$k = \frac{1}{LT} \sum_{t=1}^T \sum_{i=1}^L k_i(t),$

and spatiotemporal correlations,

$$C(\delta_i, \delta_t) = \langle k_i(t-\delta_t) k_{i+\delta_i}(t) \rangle - \langle k_i(t-\delta_t)\rangle \langle k_{i+\delta_i}(t)\rangle,$$

where the averages are over quantum trajectories. A single realization may display large, contiguous regions of high and low local measurement activity, akin to dynamical phase separation or intermittency—an archetype of space-time heterogeneity reminiscent of glassy behavior (Cech et al., 2024).

3. Large Deviation Theory and Dynamical Phase Transitions

The distribution of dynamical observables (e.g., total trajectory activity $K$) follows a large deviation principle: $P(K \approx kT) \sim \exp[-T I(k)],$ where $I(k)$ is the rate function. The moment-generating function is governed by a tilted quantum channel: $$\mathcal{E}_s[\rho] = \sum_{\vec{k}} e^{-s |\vec{k}|} K_{\vec{k}} \rho K_{\vec{k}}^\dagger,$$ with the scaled cumulant-generating function $$\theta(s) = \lim_{T\to\infty} (1/T) \ln \langle e^{-sK}\rangle$$ associated with the largest eigenvalue of $\mathcal{E}_s$.

A central result is the existence of a first-order dynamical phase transition in trajectory space, signaled by a cusp in $\theta(s)$ and non-convexity in $I(k)$. This coexistence of low- and high-activity phases yields bimodal distributions of dynamical observables and strong finite-size scaling in variances, paralleling dynamical phase transitions in classical kinetically constrained models (Cech et al., 2024).

4. Quantum Trajectories, Information Flow, and Learnability Transitions

Monitored evolution implements a quantum-classical channel, mapping an initial state $\rho$ to a set of measurement records $|m\rangle_C$ with outcome probabilities: $$\rho \rightarrow \sum_m K_m \rho K_m^\dagger \otimes |m\rangle\langle m|_C.$$ The "informational power" $W(\Pi)$ of the associated POVM quantifies how much information about the initial state can be extracted from the measurement record. $W(\Pi)$ undergoes a sharp transition at the measurement-induced phase transition (MIPT). In the entangling (low-$p$) phase, $W(\Pi) \to 0$ exponentially in system size—information is effectively hidden; in the disentangling (high-$p$) phase, $W(\Pi)$ becomes $O(1)$, permitting efficient classical shadow tomography. The sample complexity of reconstructing observables or state fidelity exhibits sharp jumps at the MIPT, establishing "learnability" as a precise operational manifestation of the dynamical phase transition (Ippoliti et al., 2023).

5. Monitored Dynamics in Specific Settings: Free Fermions, Topology, and Many-Body Localization

Monitored Free Fermions: For free-fermion systems with local density monitoring, the dynamics map onto a nonlinear sigma model (NLSM) in $d+1$ dimensions, with competition between unitary hopping ($J$) and measurement ($\gamma$). The purification/localization time $T^*(L,\gamma)$ grows distinctively in different phases: (delocalized/critical/localized) depending on $\gamma$ relative to the critical point $\gamma_c$. The scaling collapse and critical exponents are quantitatively determined; in $d=2$, the measurement-induced transition aligns with the Anderson localization transition (Poboiko et al., 1 Dec 2025). Universal stochastic (Fokker-Planck) equations describe the time evolution of density-matrix spectra, yielding parity effects in purification and universal entropy fluctuations tied to Altland-Zirnbauer symmetry class (Xiao et al., 2024).

Monitored Topological Phases: Monitored dynamics of free (Majorana or complex) fermions are classified under the ten-fold Altland-Zirnbauer symmetry scheme, with nontrivial bulk/topological invariants in spacetime dimension $d+1$. The corresponding nonlinear sigma models admit topological terms ($\Theta$, WZW), leading to bulk-boundary correspondence in the Lyapunov spectrum: topologically protected edge modes produce algebraic rather than exponential purification at boundaries (Xiao et al., 2024, Pan et al., 2024). Dynamical protocols can realize and braid Majorana-like modes protected by monitored topological order.

Prethermal Many-Body Localization: In MBL/Floquet systems, rare projective measurements destabilize the prethermal plateau, enhancing entanglement and inducing emergent classical structure in the measurement records. Analysis via PCA and mutual information directly on the outcome dataset uncovers the transition from nonergodic to thermalizing regimes, offering experimentally accessible diagnostics via the classical record (Sun et al., 14 Mar 2025).

6. Experimental Realizations and Large-Scale Simulation

The monitored collision protocol—preparation, entangling gate, ancilla measurement, reset—is directly implementable in Rydberg atom arrays and digital quantum processors. Trotterized quantum circuits realize the required kinetic constraints and mid-circuit measurements, with no post-selection overhead for trajectory statistics. Scaling properties of dynamical phase transitions, including the sharpening of response and emergence of tie-lines, are accessible in current quantum hardware (Cech et al., 2024).

7. Outlook and Extensions

Monitored quantum dynamics constitutes a universal framework bridging out-of-equilibrium statistical mechanics, quantum information, and experimental quantum simulation. The interplay between measurement and unitary evolution gives rise to rich nonequilibrium phenomena: dynamical phase coexistence, hidden glassy/heterogeneous dynamics, learnability transitions, and topologically protected slow modes. The large deviation approach, together with informational and topological diagnostics, provides organizing principles applicable across platforms, including spin chains, bosons, fermions, and systems with symmetry and disorder.

Future directions include: (i) generalizing dynamical large-deviation frameworks to more complex measurement protocols and higher dimensions, (ii) probing the impact of finite-rate post-selection and partial measurement, (iii) establishing the influence of classical computational complexity on learnability and decoding, and (iv) exploring the correspondence between dynamical phase coexistence in quantum and classical glassy systems.

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References:

• (Cech et al., 2024) Space-time correlations in monitored kinetically constrained discrete-time quantum dynamics
• (Ippoliti et al., 2023) Learnability transitions in monitored quantum dynamics via eavesdropper's classical shadows
• (Poboiko et al., 1 Dec 2025) Quantum dynamics of monitored free fermions
• (Xiao et al., 2024) Universal Stochastic Equations of Monitored Quantum Dynamics
• (Sun et al., 14 Mar 2025) Probing prethermal nonergodicity through measurement outcomes of monitored quantum dynamics
• (Xiao et al., 2024) Topology of Monitored Quantum Dynamics
• (Pan et al., 2024) Topological Modes in Monitored Quantum Dynamics
• (Li et al., 23 Jun 2025) Emergent deterministic entanglement dynamics in monitored infinite-range bosonic systems