Recent advances in quantum simulation have enabled the discovery of novel forms of universality in quantum many-body dynamics. In this Review, we discuss deep thermalization and Hilbert space ergodicity--two recently discovered phenomena characterized by the emergence of distributions of quantum states that are "maximally random" in a precise sense. They offer a new perspective on how irreversible statistical mechanics arises from reversible unitary quantum dynamics, going beyond conventional theories of quantum thermalization and equilibration. We review a unifying framework, rooted in quantum information theory and principles of maximum entropy, that explains the different forms of ergodicity which arise under various physical constraints. We discuss open questions and active research directions, including generalizations beyond ensembles of pure quantum states, phase transitions in deep thermalization tied to deeper forms of ergodicity-breaking, connections to broader topics in quantum thermalization and ergodicity, and applications in quantum information science, such as for benchmarking or tomography.
These lectures advocate the idea that quantum entanglement provides a unifying foundation for both statistical physics and high-energy interactions. I argue that, at sufficiently long times or high energies, most quantum systems approach a Maximal Entanglement Limit (MEL) in which phases of quantum states become unobservable, reduced density matrices acquire a thermal form, and probabilistic descriptions emerge without invoking ergodicity or classical randomness. Within this framework, the emergence of probabilistic parton model, thermalization in the break-up of confining strings and in high-energy collisions, and the universal small-
x behavior of structure functions arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space. Abstract Published by the Jagiellonian University 2026 authors
This paper explores the application of high-order quantum entanglement to statistical physics modeling, aiming to understand and enhance the simulation of quantum states. We propose a novel conceptual framework, utilizing the 'statistical' interpretation of entanglement, to drive the evolution of entangled states. This approach offers a potential pathway to more accurate and insightful quantum state simulations, particularly for complex quantum information processing. The core mechanism involves introducing a statistical element into the entanglement process, resulting in a refined understanding of entangled dynamics. This work seeks to advance the field of quantum information and quantum state simulation by providing a new perspective on the challenges inherent in these areas.
The ETH represents a breakthrough in many-body physics since it links thermalization of physical observables with the applicability of RMT. This framework is widely believed to hold for an overwhelming majority of physical systems, though exceptions, where conventional ETH is violated, include integrability, single-particle chaos, many-body localization, many-body scars, and Hilbert-space fragmentation. However, the mechanism of the ETH breakdown remains elusive. In this thesis, we introduce a novel scenario in many-body quantum systems, dubbed
itfadingergodicity regime, which establishes a link between the conventional ETH and non-ergodic behavior. This framework predicts an exponent,
eta, governing eigenstate fluctuations in the matrix elements of local observables. As a consequence, slow dynamics emerge as the perturbations become vanishingly small, and the Thouless energy matches the Heisenberg scale at the critical point. We conjecture this scenario to be relevant for the description of many-body systems, where the divergent relaxation time is described by the Fermi golden rule. We provide numerical and analytical arguments for its validity in the quantum sun model and related RMT models. We continue our discussion by exploring the universality of the fading ergodicity scenario. Remarkably, we show that fading ergodicity leads to a maximally divergent peak in the fidelity susceptibility, establishing the onset of
itmaximalchaos at the critical point. Furthermore, we extend this framework to finite energy density, with both fading ergodicity and maximal chaos following the many-body mobility edge to high precision. Finally, we present a scaling theory of many-body ergodicity breaking, which exhibits a critical exponent
nu=1 at the critical point, characterizing the universality class of fading ergodicity. However, in contrast to the well established Anderson localization transition, we argue that the one-parameter scaling theory is insufficient.
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024