Quantum evolution is generally expected to drive a quantum many-body system toward equilibrium. This expectation is often justified by the Hilbert-space ergodicity of generic quantum dynamics, namely, the idea that pure-state evolution explores Hilbert space uniformly up to physical constraints. Such a statement can be made rigorous by requiring the associated state ensemble to form the Haar-random ensemble, or its more structured generalization, the Scrooge ensemble. In this Letter, we report the emergence of exact Hilbert-space ergodicity in a continuously monitored quantum many-body system. For any target density matrix $\sigma$, we construct a continuously monitored system for which we rigorously prove that the Scrooge ensemble of $\sigma$ is the unique late-time equilibrium distribution of quantum trajectories. Remarkably, this requires only that the jump operators in the monitoring form a deformed unitary 1-design, a seemingly much weaker condition than full ergodicity. We numerically demonstrate our predictions by simulating continuously monitored systems whose equilibrium states are thermal states. Our results establish a rigorous mechanism for the emergence of Hilbert-space ergodicity and provide a practical route for its investigation on quantum devices.
We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the ``chord Hilbert space''construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-\lambda}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.
Quantum resource theories characterize distinct forms of nonclassicality in many-body quantum states, raising the question of whether these resources evolve independently under generic ergodic dynamics. Considering diagnostics quadratic in the state, we show that the dynamics of different resource measures become mutually interdependent and are governed by a few common degrees of freedom. For Haar-random circuits, the purity together with a single resource witness suffices to reconstruct the remaining resource measures, as we demonstrate for coherence, various asymmetries, and number entropies. The same relations hold, to a good approximation, under chaotic Floquet and continuous-time Hamiltonian dynamics, establishing that this dynamical resource equivalence extends beyond random circuits. We further derive an exact coherence--imaginarity relation, remarkably accurate also beyond Haar-random circuits. Our results reveal an emergent simplification of many-body dynamics, in which sufficiently strong scrambling reduces seemingly distinct quantum resources to a few common dynamical degrees of freedom.
Sreemayee Aditya, X. Turkeshi, P. Sierant· 0 citations
Projected ensembles---the collections of conditional pure states induced on a system by measuring an entangled environment---have become central objects in quantum science, underlying deep thermalization, quantum state designs, the emergence of classicality, and exhibiting interesting phase transitions. Here we develop a general and exact theory of their dynamics, yielding a systematic approach to their dynamics and equilibrium and non-equilibrium stationary states. We derive an exact continuity equation on quantum state space for the projected ensemble, together with microscopic kinetic equations and analytical expressions for the probability flux and source terms generated by the system-environment interaction. The resulting dynamics admits a classical representation: isolated systems obey Hamiltonian transport and Liouville's theorem, while open systems are described by a kinetic theory of probability transport. The stationary continuity equation provides a general characterization of equilibrium and non-equilibrium stationary projected ensembles, which can be analyzed through the method of characteristics. The theory provides an analytical framework for studying the emergence, structure, and timescales of equilibrium and non-equilibrium stationary projected ensembles, with applications ranging from deep thermalization and the emergence of classicality to random quantum-state generation and quantum device benchmarking.
Hilbert-space ergodicity (HSE) provides a new framework for studying thermalization in driven quantum systems, complementing the eigenstate thermalization hypothesis, which is restricted to static systems. This ergodicity is hierarchical: by quantifying how randomly the dynamics explores the Hilbert space, one obtains a family of levels termed $k$-HSE. While HSE has been observed at the lowest and highest levels, finite-order HSE dynamics remains largely unexplored due to the difficulty of constructing such drives. Here, we explore this intermediate regime and uncover its distinctive physics. We first propose and prove that a family of $m$-tone drives on qubits realizes $k$-HSE up to $k = 2m{-}3$, with drive parameters determined at $O(k)$ cost. Using a single nitrogen-vacancy center in diamond, we verify this design by showing that a 3-tone drive realizes 3-HSE, with fourth-order statistics depending on the initial state. Further in-depth theoretical analysis shows that this initial-state dependence is generic across drives, demonstrating the possibility of recovering the initial state from higher-order statistics even when the dynamics is ergodic. Our work broadens the study of quantum ergodicity and reveals intriguing physics within its hierarchy.
We introduce a quantum stochastic resetting protocol with uniform memory, in which each resetting event returns the system to a state visited at a time chosen uniformly from its entire history. The resulting dynamics is nonunitary, non-Markovian and a direct quantum generalization of the classical preferential relocation model. Working in the energy eigenbasis, we derive the exact evolution of every density-matrix element for an arbitrary time-independent Hamiltonian and show that the Hamiltonian enters the dynamics only through the corresponding Bohr frequencies. This leads to a natural distinction between two classes of quantum systems: gapped and gapless. In \emph{gapped systems} (systems with a discrete energy spectrum), while the diagonal elements remain unchanged, the off-diagonal elements of the density matrix in the energy eigenbasis decay algebraically with a continuously varying exponent and with an amplitude that oscillates periodically in $\log t$. The system therefore approaches a stationary state that is independent of the resetting rate and retains a strong memory of the initial state. In \emph{gapless systems} (systems with a continuous energy spectrum), arbitrarily small Bohr frequencies prevent stationarity. Instead, the position distribution spreads on the universal (ultra-slow) scale $\log(rt)/r$, independently of the initial state and of the details of the Hamiltonian. We illustrate these results with a two-level system, a harmonic oscillator, and a free quantum particle, and contrast them with their classical counterparts.
G. Mauro, Manas Kulkarni, S. Majumdar· 0 citations
Stochastic quantization defines a Euclidean quantum field theory as the equilibrium of a fictitious-time Langevin dynamics, which reaches the Gibbs measure asymptotically. We show that this quantization can be formulated as a finite-time stochastic optimal control problem. A tractable reference process, naturally supplied by the free theory when available, provides an Ornstein--Uhlenbeck dynamics, while the full interaction enters as a reference-corrected terminal cost. The optimal control is a Doob-transform force that steers the path-reweighted terminal ensemble to the target at a prescribed time and for a given noise amplitude. A neural network learns the residual control, realizing this optimal stochastic quantization (OSQ). Because the path weights are exact, imperfect training increases the variance of estimators but does not introduce model bias. On multimodal potentials we recover all modes at finite time and find that the noise amplitude sets a practical diffusion-horizon window. In two-dimensional lattice scalar $\phi^4$ theory we recover observables from hybrid Monte Carlo simulations near the critical point. Quantization is thereby formulated as control rather than equilibration.