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arXiv:2408.16974 · Physical Review Letters

Universal Stochastic Equations of Monitored Quantum Dynamics

Zhenyu Xiao, Tomi Ohtsuki, Kohei Kawabata

Published Apr 7, 2025• 9 citations
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Abstract

We investigate the monitored quantum dynamics of Gaussian mixed states and derive the universal Fokker-Planck equations that govern the stochastic time evolution of entire density-matrix spectra, obtaining their exact solutions. From these equations, we reveal an even-odd effect in purification dynamics: whereas entropy exhibits exponential decay for an even number N of complex fermions, algebraic decay with divergent purification time occurs for odd N as a manifestation of dynamical criticality. Additionally, we identify the universal fluctuations of entropy in the chaotic regime, serving as a nonunitary counterpart of the universal conductance fluctuations in mesoscopic electronic transport phenomena. Furthermore, we elucidate and classify the universality classes of nonunitary quantum dynamics based on fundamental symmetry. We also validate the universality of these analytical results through extensive numerical simulations across different types of models.

DOI
10.1103/physrevlett.134.140401

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We investigate the monitored quantum dynamics of Gaussian mixed states and derive the universal Fokker-Planck equations that govern the stochastic time evolution of entire density-matrix spectra, obtaining their exact solutions. From these equations, we reveal an even-odd effect in purification dynamics: whereas entropy exhibits exponential decay for an even number N of complex fermions, algebraic decay with divergent purification time occurs for odd N as a manifestation of dynamical criticality. Additionally, we identify the universal fluctuations of entropy in the chaotic regime, serving as a nonunitary counterpart of the universal conductance fluctuations in mesoscopic electronic transport phenomena. Furthermore, we elucidate and classify the universality classes of nonunitary quantum dynamics based on fundamental symmetry. We also validate the universality of these analytical results through extensive numerical simulations across different types of models.
We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; ii) weak continuous-time monitoring that induces a Dyson brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus's operator space, focusing in particular on the unitary and orthogonal cases, respectively β=2β=2β=2 and β=1β=1β=1, with βββ the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any βββ, thanks to the mapping of the Fokker-Planck evolution of eigenvalues onto the Calogero-Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
Abstract We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement-induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: (i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; (ii) weak continuous-time monitoring that induces a Dyson Brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus’s operator space, focusing in particular on the unitary and orthogonal cases, respectively β = 2 and β = 1 , with β the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any β , thanks to the mapping of the Fokker–Planck evolution of eigenvalues onto the Calogero–Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.
We investigate the purification dynamics of a single qubit under continuous in time monitoring. By employing a collisional model framework where the system interacts sequentially with ancillary qubits, we describe the conditioned evolution of the density matrix through a stochastic master equation. We show that for initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion dominated regime to a measurement dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations, providing a robust theoretical benchmark for the study of information extraction in monitored quantum systems.
We consider the universal aspects of two problems: (i) the singular value structure of a product M t = m t m t − 1 … m 1 of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices m i , representing time steps in the evolution, but the m i are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.
We consider the evolution of continuously measured many-body chaotic quantum systems.Focusing on the dynamics of state purification, we analytically describe the limits of strong and weak measurement rate, where the latter case is challenging in that monitoring up to time scales exponentially long in the numbers of particles is required.We complement the analysis of the limiting regimes with the construction of an effective replica theory providing information on the stability and the symmetries of the respective phases.The analytical results are tested by comparison to exact diagonalization.