Abstract I employ random-matrix methods to set up and solve statistical models of noisy nonunitary dynamics that appear in the context of monitored quantum systems. The models cover a range of scenarios combining random dynamics and measurements of variable strength of one or several qubits. The combined dynamics drive the system into states whose statistics reflect the competition of randomizing unitary evolution and the measurement-induced backaction collapsing the state. These effects are mediated by entanglement, as I describe in detail by analytical results. For the paradigmatic case of monitoring via a single designated qubit, this reveals a simple statistical mechanism, in which the monitoring conditions the state of the monitored qubit, which then imposes statistical constraints on the remaining quantities of the system. For the case of monitoring several qubits with prescribed strength, the developed formalism allows one to set up the statistical description and solve it numerically. Finally, I also compare the analytical results to the monitored dynamics of a quantum kicked top, revealing two regimes where the statistical model either describes the full stationary dynamics, or resolves time scales during particular parts of the evolution.
We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter-Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker-Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.
Kinetically constrained models have been widely studied in the context of glass formers and non-equilibrium statistical mechanics. Although their simple local rules often result in structureless static properties, their dynamics exhibit intricate emergent phenomena. In this work, we investigate monitored quantum circuit models that interpolate between classical stochastic and unitary quantum dynamics. For any finite measurement strength, the measurement records provide an experimentally accessible probe of the emergence of dynamical phases. By interpreting space-time resolved records as microstates of a fictitious 1+1D spin system, we employ thermodynamic concepts that allow us to investigate the dynamical coexistence between an active and inactive phase. We combine insights from classical stochastic dynamics and numerical simulations of monitored quantum dynamics to investigate different signatures of this dynamical phase coexistence as the measurement strength is varied. Our results shed light on the persistence of dynamical phase coexistence in the quantum regime, offering insights into future experimental studies of complex many-body dynamics in quantum simulators.
Abstract We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter–Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker–Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.
The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when
N increases, with two plateaus corresponding to integers, where
Fn is the first detected return probability at the
nth detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability
Fn decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of
Fn. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.
We study the dynamics of Loschmidt echoes in noisy quantum many-body systems without conservation laws. We first show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging. We then analyze this quantity in two complementary ways, revealing universal dynamical behavior. First, we develop a heuristic picture for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth, which is valid at any dissipation strength. We assert that the Loschmidt echo has two dynamical regimes depending on the time t and the strength of the noise p : Gaussian decay for p t ≪ 1 and exponential decay (with a noise-independent decay rate) for p t ≫ 1 . Lastly, we rigorously prove all our results for a solvable chaotic many-body quantum circuit, the dissipative random phase model—thus providing exact insight into dissipative quantum chaos.