Exact Hilbert-Space Ergodicity in Quantum Dynamics

• Exact Hilbert-space ergodicity is defined as the complete matching of all temporal tensor-power moments with those of the Haar (or Scrooge) ensemble.
• It establishes a stronger notion than the eigenstate thermalization hypothesis by demanding full state-space exploration through dynamic evolution.
• Explicit constructions like generalized Fibonacci and Thue-Morse drives achieve this ergodicity, while symmetry and locality introduce fragmentation that can obstruct it.

Exact Hilbert-space ergodicity is a strong dynamical notion of quantum ergodicity in which the long-time temporal ensemble generated by a quantum evolution coincides exactly with a reference uniform ensemble on projective Hilbert space. In the closed-system setting this reference ensemble is typically the Haar ensemble of pure states; in continuously monitored settings it can be replaced by the Scrooge ensemble associated with a prescribed full-rank density matrix. The concept is stronger than the eigenstate thermalization hypothesis because it requires uniform exploration of the entire available state space by the dynamics itself, rather than randomness only in matrix elements or in typical eigenstates. Recent work has formalized this notion through moment equalities, established rigorous no-go theorems for broad classes of dynamics, constructed exact ergodic drives, and identified symmetry- and locality-based obstructions that force fragmentation into dynamically disconnected sectors (Pilatowsky-Cameo et al., 2023, Pilatowsky-Cameo et al., 2024, Wu et al., 27 Jun 2026, Sohal et al., 26 Nov 2025).

1. Formal notion and moment structure

For a finite $d$-dimensional Hilbert space $\mathcal H$, with $\ket{\psi(t)}=U(t)\ket{\psi(0)}$, complete Hilbert-space ergodicity (CHSE) is defined by the requirement that for every initial state $\ket{\psi(0)}$ and every continuous, or even merely integrable, function $f(\ket{\psi}\bra{\psi})$ on the pure-state manifold,

$$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$

Equivalently, one may require agreement of all tensor-power moments: $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$ with

$$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$

This identifies CHSE with exact convergence of the temporal state ensemble to an infinite-order state design (Pilatowsky-Cameo et al., 2023).

A parallel formulation for driven systems introduces the temporal distribution

$$\mu_{\rm time}=\lim_{T\to\infty}\frac1T\int_0^T \delta_{\psi(t)}\,dt,$$

and demands exact equality with the Haar distribution on pure states. The corresponding unitary version, termed complete unitary ergodicity (CUE), replaces states by channels $\mathcal U(t)[\cdot]=U(t)(\cdot)U(t)^\dagger$ and requires equality of all channel moments

$\mathcal H$0

with their Haar counterparts. In the state case the Haar moment admits the Schur-form expression

$\mathcal H$1

where $\mathcal H$2 projects onto the symmetric subspace (Pilatowsky-Cameo et al., 2024).

2. Relation to ETH, designs, and dynamical randomness

The literature treats exact Hilbert-space ergodicity as a dynamical analogue of the intuitive classical notion of ergodicity. It is explicitly distinguished from Berry’s conjecture and from the eigenstate thermalization hypothesis (ETH): ETH is formulated in terms of static eigenstates and local observables, whereas CHSE is a statement about the full long-time distribution of the evolving state and makes no reference to any stationary basis. Under CHSE, transition amplitudes also acquire Haar statistics; in particular, for any fixed $\mathcal H$3,

$\mathcal H$4

The same framework implies that finite-$\mathcal H$5 moment matching is equivalent to the temporal ensemble forming a state or unitary $\mathcal H$6-design (Pilatowsky-Cameo et al., 2023).

Operationally, driven-system work reformulates exact HSE and exact CUE through frame potentials. For states,

$\mathcal H$7

and exact moment matching is equivalent to $\mathcal H$8 for all $\mathcal H$9. An analogous unitary frame potential $\ket{\psi(t)}=U(t)\ket{\psi(0)}$0 characterizes exact CUE. This places exact Hilbert-space ergodicity within the same formal hierarchy as exact quantum design generation, but with a temporal ensemble rather than an externally sampled one (Pilatowsky-Cameo et al., 2024).

3. No-go theorems and structural obstructions

Exact Hilbert-space ergodicity is impossible for time-independent and time-periodic quantum dynamics. If the evolution possesses energy or quasienergy eigenstates, the populations in those states are conserved, and for $\ket{\psi(t)}=U(t)\ket{\psi(0)}$1 one obtains a nonzero lower bound

$\ket{\psi(t)}=U(t)\ket{\psi(0)}$2

Accordingly, CHSE cannot hold for static Hamiltonians or Floquet systems in the strict sense, even when those systems are otherwise chaotic in more conventional senses (Pilatowsky-Cameo et al., 2023).

For general driven systems, exact ergodicity imposes quantitative lower bounds on drive complexity. For single-frequency drives with finite action

$\ket{\psi(t)}=U(t)\ket{\psi(0)}$3

Theorem 1 in the driven-systems analysis states that CHSE and CUE are impossible. More generally, if an $\ket{\psi(t)}=U(t)\ket{\psi(0)}$4-tone quasiperiodic drive admits a Floquet-type decomposition

$\ket{\psi(t)}=U(t)\ket{\psi(0)}$5

with $\ket{\psi(t)}=U(t)\ket{\psi(0)}$6 quasiperiodic in $\ket{\psi(t)}=U(t)\ket{\psi(0)}$7 angles and $\ket{\psi(t)}=U(t)\ket{\psi(0)}$8 having $\ket{\psi(t)}=U(t)\ket{\psi(0)}$9 distinct quasienergies, then exact state ergodicity requires $\ket{\psi(0)}$0, while exact unitary ergodicity requires $\ket{\psi(0)}$1. By contrast, finite-order pseudo-randomness remains accessible: a single-frequency Floquet drive may realize a finite $\ket{\psi(0)}$2-design if the period or Hamiltonian action grows sufficiently with $\ket{\psi(0)}$3 and $\ket{\psi(0)}$4, but exact all-moment ergodicity does not follow (Pilatowsky-Cameo et al., 2024).

A separate obstruction arises from locality and exact higher-form symmetries. In a finite-dimensional local lattice system with an exact $\ket{\psi(0)}$5 $\ket{\psi(0)}$6-form symmetry, the Hilbert space splits into exponentially many dynamically disconnected Krylov subspaces. For a two-dimensional lattice with a layering of $\ket{\psi(0)}$7 non-overlapping loops, the number of sectors satisfies

$\ket{\psi(0)}$8

These sectors are not distinguished by symmetry quantum numbers alone; rather, they are characterized by emergent integrals of motion $\ket{\psi(0)}$9 associated with open paths between neighboring extremal loops. The result is a symmetry-based obstruction to exact ergodicity that depends only on locality, finite on-site Hilbert-space dimension, and exact higher-form symmetry, not on microscopic Hamiltonian details (Sohal et al., 26 Nov 2025).

4. Exact constructions in closed driven systems

The first rigorous closed-system construction of CHSE in this corpus is the family of generalized Fibonacci drives. Starting from two unitaries $f(\ket{\psi}\bra{\psi})$0, one defines the binary words

$f(\ket{\psi}\bra{\psi})$1

whose infinite limit has minimal aperiodic symbolic complexity $f(\ket{\psi}\bra{\psi})$2. Applying the corresponding letters as a deterministic discrete-time drive yields a temporal averaging channel whose infinite-time limit equals the Haar-averaging channel in every tensor-power sector for almost all pairs $f(\ket{\psi}\bra{\psi})$3. Hence generalized Fibonacci drives of any order $f(\ket{\psi}\bra{\psi})$4 exhibit CHSE almost surely in dimension $f(\ket{\psi}\bra{\psi})$5 (Pilatowsky-Cameo et al., 2023).

The driven-systems design framework complements this result with explicit geometric constructions. For a single qubit, a two-tone quasiperiodic drive can be engineered so that $f(\ket{\psi}\bra{\psi})$6 exactly parametrizes $f(\ket{\psi}\bra{\psi})$7 through a measure-preserving Euler-angle map. For general qudits, an explicit Hurwitz-angle parametrization yields an $f(\ket{\psi}\bra{\psi})$8-tone family whose $f(\ket{\psi}\bra{\psi})$9 exactly covers $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$0. These constructions are sufficient for exact state and unitary ergodicity, in contrast to the obstruction theorems for low-tone drives with quasienergy structure (Pilatowsky-Cameo et al., 2024).

A different exact construction appears in quantum morphic drives. For the Thue-Morse drive on a qubit, with generic $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$1, the temporal unitary ensemble forms a unitary $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$2-design for every $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$3 in the $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$4 limit, implying exact CUE and therefore CHSE. At the same time, the dynamics exhibits emergent approximate discrete time-translation symmetries over arbitrarily long but finite intervals, producing long-lived plateaus in the distance to Haar. The plateau durations satisfy

$$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$5

so the approach to exact ergodicity is critically slow. The same work identifies regular CHSE, critically slow CHSE, and asymptotically Floquet behavior as distinct classes among morphic drives generated by short substitution rules (Pilatowsky-Cameo et al., 10 Feb 2025).

5. Open-system generalization: continuous monitoring and the Scrooge ensemble

Continuously monitored many-body systems provide an exact open-system mechanism for Hilbert-space ergodicity. In that setting, ergodicity is formulated measure-theoretically: for almost every initial state, time averages of observables along stochastic quantum trajectories converge to expectations over a unique invariant ensemble on $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$6. The invariant ensemble need not be Haar. Given a full-rank target density matrix $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$7, the relevant stationary ensemble is the Scrooge ensemble $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$8, defined as the maximum-entropy ensemble of pure states whose average reproduces $$\lim_{T\to\infty}\frac1T\int_0^T dt\;f\!\bigl(\ket{\psi(t)}\bra{\psi(t)}\bigr) = \int_{\mathrm{Haar}} d\phi\; f\!\bigl(\ket{\phi}\bra{\phi}\bigr).$$9 (Wu et al., 27 Jun 2026).

The monitored dynamics is constructed using jump operators

$$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$0

where the $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$1 satisfy a deformed unitary $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$2-design condition and a phase-neutrality condition. The main theorem states that the Scrooge distribution is the unique stationary distribution and that for any initial pure-state distribution $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$3,

$$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$4

A notable feature is that exact ergodicity emerges here from a deformed unitary $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$5-design, without requiring higher $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$6-designs or chaotic Hamiltonian evolution. Numerical demonstrations in the same work use thermal target states $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$7 and observe convergence of trajectory statistics to the analytic Scrooge distribution (Wu et al., 27 Jun 2026).

6. Fragmentation, subspace ergodicity, and diagnostics

When full exact Hilbert-space ergodicity fails because the Hilbert space decomposes into invariant dynamical blocks, the appropriate refinement is Hilbert subspace ergodicity. If

$$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$8

then one can replace the full Haar measure by the Haar measure on a given $$\rho_T^{(k)}=\frac1T\int_0^T dt\;(\ket{\psi(t)}\bra{\psi(t)})^{\otimes k}, \qquad \rho_{\rm Haar}^{(k)}=\int_{\rm Haar} d\phi\;(\ket{\phi}\bra{\phi})^{\otimes k},$$9. Complete Hilbert subspace ergodicity (CHSSE) requires that for every initial state in $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$0, the time-ensemble moments converge to the Haar moments of that subspace. This framework captures the effect of exact symmetries, quantum many-body scars, and Hilbert-space fragmentation: full CHSE is destroyed, but exact uniform sampling may still occur within each invariant block (Logaric et al., 2024).

Concrete many-body examples illustrate this distinction. In the detuned Rydberg-Ising chain mapped to a folded XXZ model, generic symmetry sectors fragment into exponentially many disjoint Krylov subspaces, and an initial product state can only explore its own $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$1. Depending on parameters, one finds an integrable regime, a Krylov-restricted thermal phase, and statistical-bubble localization; full Hilbert-space ergodicity occurs only in fully connected sectors such as magnon-only or hole-only sectors (Yang et al., 2024). In multicomponent Hubbard models, generalized $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$2-pairing states and related towers define exact eigenstates and exponentially large integrable subsectors in which the Hubbard Hamiltonian reduces effectively to a free-fermion model; this produces weak ETH violation and partial integrability, ruling out exact Hilbert-space ergodicity in the full space (Nakagawa et al., 2022).

Several diagnostics make these structures explicit. Spectral graph theory maps basis states to vertices and nonzero Hamiltonian matrix elements to edges. Exact fragmentation corresponds to multiple disconnected graph components, equivalently to multiple zero modes of the graph Laplacian $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$3. Exact ergodicity, up to trivial symmetry sectors, corresponds to a connected graph with a unique zero eigenvalue and spectral gap $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$4. Nearly fragmented systems are formally connected but have $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$5, while Fiedler vectors and modularity spectra reveal weakly coupled communities (Rutkowski et al., 18 May 2026). In a distinct experimental direction, probe-qubit fluctuation diagnostics relate the long-time variance $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$6, the decay rate $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$7, and the effective bath dimension $$\Delta_\infty^{(k)}=\tfrac12\left\|\lim_{T\to\infty}\rho_T^{(k)}-\rho_{\rm Haar}^{(k)}\right\|_1=0 \quad\text{for all }k.$$8 through an infinite-temperature fluctuation-dissipation theorem, providing an operational route to infer how much Hilbert space is effectively accessed in a chaotic device (Nation et al., 2019).

Taken together, these results delimit a precise modern meaning of exact Hilbert-space ergodicity. It is an all-moment, ensemble-level statement of uniform exploration; it is stronger than ETH; it is impossible for static and Floquet dynamics in the strict sense; it can nevertheless be achieved exactly by specially structured aperiodic drives and by continuous monitoring; and it is fundamentally obstructed by invariant sector decompositions produced by symmetries, locality, and fragmentation. Within fragmented settings, the appropriate exact notion is often not full-space ergodicity but exact ergodicity relative to a dynamically closed subspace (Logaric et al., 2024).