Exact Hilbert-space ergodicity from continuous monitoring

• URL: http://arxiv.org/abs/2606.29042v2
• Date: Tue, 30 Jun 2026 08:11:34 GMT
• Title: Exact Hilbert-space ergodicity from continuous monitoring
• Abstract summary: We report the emergence of exact Hilbert-space ergodicity in a continuously monitored quantum many-body system.<n>Our results establish a rigorous mechanism for the emergence of Hilbert-space ergodicity and provide a practical route for its investigation on quantum devices.
• Score: 13.060111321441296
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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