Exact Hilbert-space ergodicity from continuous monitoring

Abstract: 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 $σ$, we construct a continuously monitored system for which we rigorously prove that the Scrooge ensemble of $σ$ 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.

• The paper establishes the main theorem that continuous quantum measurements drive pure states to explore Hilbert space uniformly according to the Scrooge ensemble.
• It derives explicit stochastic Schrödinger and Fokker–Planck equations to describe the drift and diffusion dynamics induced by continuous monitoring.
• The work highlights practical implications for quantum control and state engineering by demonstrating exponential convergence to a unique stationary distribution.

Exact Ergodicity in Hilbert Space from Continuous Quantum Monitoring

Introduction

This work analyzes the ergodicity properties of quantum trajectories under continuous generalized measurement, specifically focusing on pure-state Stochastic Schrödinger equations (SSEs) with jump operators engineered from unitary ensembles. The authors rigorously prove that the continuous monitoring protocol drives the pure quantum state to explore the complex projective Hilbert space $\mathbb{CP}^{D-1}$ exactly according to the so-called Scrooge ensemble, with the unique stationary distribution fully determined by the average density matrix $\sigma$ imposed by the measurement protocol.

Scrooge Ensemble Construction and Probabilistic Structure

The Scrooge ensemble $\mathcal{E}_{\mathrm{sc}(\sigma)}$ emerges as a central object, defined via a re-weighted Haar measure on pure states whose average is a fixed full-rank density matrix $\sigma$. Explicit parameterization in the eigenbasis of $\sigma$ enables tractable analysis: decomposing pure states into amplitudes $y_i$ and relative phases $\varphi_i$, the resulting probability density on the projective simplex is shown to be proportional to $H(\bm y)^{-(D+1)}$ with $H(\bm y) = \sum_i y_i/\lambda_i$. The invariance of the phase distribution is secured by the protocol's symmetry.

Derived Stochastic Schrödinger and Fokker–Planck Dynamics

Using explicit Kraus operators for infinitesimal measurement, the authors derive the exact SSE for single and multiple measurement channels. Projecting to coordinates on the pure-state manifold, they arrive at coupled Langevin equations for the amplitude and phase variables, with deterministic (drift) and stochastic (diffusion) components determined by $\sigma$ and the measurement ensemble.

From these equations, the associated Fokker–Planck equation for the evolving probability density $\sigma$0 on $\sigma$1 is constructed. Analytical calculation of the drift vector and diffusion tensor identifies their dependence solely on the ensemble statistics and the prescribed average state.

Main Theorem: Stationarity, Uniqueness, and Convergence

The core theoretical result—proved in detail—is the uniqueness of the stationary distribution for the SSE under continuous measurement, and its coincidence with the Scrooge ensemble. Stationarity is established by demonstrating vanishing of the probability current in the projective amplitude coordinates for $\sigma$2.

The uniqueness and global convergence are guaranteed by a combination of mathematical arguments:

• Uniform ellipticity: The diffusion tensor is bounded below everywhere in projective Hilbert space, ensuring the process is mixing with respect to the Fubini–Study metric.
• $\sigma$3-Regularity: All transition probability measures are mutually absolutely continuous and admit smooth, strictly positive densities, following from parabolic regularity and compactness.
• Doob’s Theorem: These properties ensure a unique stationary measure attracts any initial distribution in total variation.

The entire proof structure is robust, handling the lower-rank case by restricting to the appropriate projective subspace.

Notable Claims and Numerical Implications

• Exact Hilbert-space ergodicity: Continuous monitoring protocols with the specified full-rank $\sigma$4 and unitary ensembles are shown to generate exact uniform ergodic exploration of pure states with respect to the Scrooge ensemble measure, not merely in the sense of time averages but in the strong sense of probability measure convergence.
• Exponential convergence: The population outside the support of $\sigma$5 decays exponentially, confining the long-term dynamics to the support of the measurement-determined ensemble.

The work does not present explicit numerical simulations, but the analytical rigor makes the stationary distribution and mixing times in principle computable, fully specified by $\sigma$6 and the spectrum of the measurement ensemble.

Theoretical and Practical Implications

The results have direct implications for quantum information science, quantum control, and foundational studies of open quantum systems:

• Quantum State Engineering: Measurement-driven protocols can prepare arbitrary Scrooge-ensemble states, constrained solely by the ability to implement the appropriate $\sigma$7 and unitary ensemble structure.
• Thermalization Without Hamiltonian Chaos: The result illustrates that continuous measurement can thermalize pure states exactly in the sense of Hilbert-space exploration, independently of any Hamiltonian dynamics.
• Markovian Open System Control: The explicit Fokker–Planck formalism suggests engineering possibilities for robust stochastic evolution in finite-dimensional open quantum systems, with uniquely controllable steady-states.
• Foundations: The rigorous ergodicity theorem forms a bridge between quantum probability, information geometry, and classical ergodic theory for Markov processes in high-dimensional state spaces.

Future Directions

This framework suggests several research directions:

• Explicit rate of convergence: Quantitative estimates (e.g., spectral gap bounds) for mixing times in the Fokker–Planck dynamics.
• Robustness to noise and imperfections: Analysis of how deviations from idealized measurement (e.g., non-unitary ensembles, finite efficiency) perturb the unique stationary measure.
• Extensions to infinite-dimensional Hilbert spaces: Generalization to cases where $\sigma$8.
• Applications to quantum simulation and randomized benchmarking: Use of exact Hilbert-space ergodicity to design measurement-based protocols for quantum computation or certification.

Conclusion

This paper provides a mathematically rigorous foundation for exact Hilbert-space ergodicity induced by continuous quantum monitoring, culminating in a unique stationary measure (the Scrooge ensemble) for the pure-state stochastic dynamics. The derived results bridge quantum measurement theory and Markov process ergodic theorems, offering new avenues for quantum state engineering and foundational studies in open quantum system dynamics (2606.29042).