Deep Thermalization in Quantum Systems

• Deep thermalization is a refined quantum equilibration process where post-measurement ensembles converge to maximally entropic, Haar-random or Scrooge distributions beyond mere local observable relaxation.
• It employs k-design properties to capture all higher-order moments of the projected ensemble, revealing deeper ergodic behavior and information scrambling in many-body systems.
• Using maximum entropy principles and physical constraints, deep thermalization provides operational benchmarks for benchmarking quantum dynamics and experimental implementations.

Deep thermalization is a refined notion of equilibration in quantum many-body systems, characterized not merely by the relaxation of local observables or reduced density matrices to thermal values, but by the universal statistics of entire ensembles of post-measurement pure (or mixed) states, termed projected ensembles, obtained by measuring the environment of a subsystem. This concept formalizes a stronger form of ergodicity: while standard quantum thermalization ensures convergence of local expectation values or density matrices, deep thermalization imposes constraints on all higher-order moments of the post-measurement ensemble, including their convergence to maximally entropic, random matrix–like distributions, subject to physical symmetries and conservation laws.

1. Definition and Formal Framework

Consider a partition of a quantum system into a finite subsystem $A$ and its large complement $B$, with the total (pure) state $|\Psi\rangle$ (or a density matrix $\rho$ in the mixed-state case). The projected ensemble (PE) is defined by performing a full projective measurement of $B$ in a local product basis $\{|z\rangle_B\}$, yielding outcome $z$ with Born probability $p(z)$ and projecting $A$ into the normalized post-measurement pure state $|\psi_z\rangle$: $$p(z) = \| (I_A \otimes \langle z|_B) |\Psi\rangle \|^2 \qquad |\psi_z\rangle = \frac{(I_A \otimes \langle z|_B)|\Psi\rangle}{\sqrt{p(z)}}$$ The projected ensemble is the set $$\mathcal{E}_{\text{PE}} = \{ p(z), |\psi_z\rangle \}$$.

Ordinary thermalization ensures that the reduced density matrix $$\rho_A = \mathrm{Tr}_B |\Psi\rangle\langle\Psi| = \sum_z p(z) |\psi_z\rangle\langle\psi_z|$$ relaxes to a thermal (e.g., maximally mixed) state. Deep thermalization extends this by demanding that the entire post-measurement ensemble approximates a maximally random ("Haar") distribution not only in its first moment but in all higher moments (i.e., it forms a unitary $k$-design for all $k$ as $|B| \to \infty$): $$M_{\text{PE}}^{(k)} = \sum_z p(z) (|\psi_z\rangle\langle\psi_z|)^{\otimes k} \quad\overset{?}{\to}\quad M_{\text{Haar}}^{(k)} = \int_{\text{Haar}} d\psi\, (|\psi\rangle\langle\psi|)^{\otimes k}$$ This condition probes the full structure of the wavefunction distribution, not just its mean, revealing a fundamentally deeper layer of thermalization and ergodicity (Shrotriya et al., 2023, Ippoliti et al., 2022).

2. Universal Ensembles and the Maximum Entropy Principle

The statistical form attained by the projected ensemble is governed by a maximum entropy principle: among all possible distributions of pure states consistent with physical constraints (e.g., energy, conserved charges, symmetries), the one maximizing the ensemble (Shannon) entropy is selected (Mark et al., 2024). The uniquely determined distribution is the "Scrooge ensemble" (or its generalizations), defined by maximizing

$$\text{Ent}(\mathcal{E}) = - D_{\rm KL}[P(\psi) \,\|\, P_{\rm Haar}(\psi)]$$

subject to $$\int P(\psi) |\psi\rangle\langle\psi| d\psi = \rho_A$$. The resulting $k$-th moment operator,

$$M_{\text{Scrooge}}^{(k)} = \int P_{\text{Scrooge}}(\psi) (|\psi\rangle\langle\psi|)^{\otimes k} d\psi,$$

generalizes the Haar measure to non-infinite temperature and arbitrary symmetries: when $\rho_A$ is maximally mixed, $M_{\text{Scrooge}}^{(k)} = M_{\text{Haar}}^{(k)}$.

Physical symmetries and conservation laws further structure these ensembles. For example, charge-conserving dynamics induce a block-diagonal form or "generalized Scrooge ensemble," with probability weights determined by the overlap of the initial state's charge distribution and the measurement basis (Chang et al., 2024). In free-integrable systems, the ensemble is determined by all local integrals of motion, leading to a deep Generalized Gibbs Ensemble (dGGE) (2207.13628).

3. Dynamical Emergence and Universality Classes

The approach to deep thermalization depends on both the microscopic dynamics and the system's conservation laws. In maximally chaotic or dual-unitary circuits, deep thermalization occurs rapidly and all moments converge to the universal (Haar or Scrooge) form (Shrotriya et al., 2023, Ippoliti et al., 2022, Bhore et al., 2023). In models possessing additional conserved quantities, such as $U(1)$ charge, the universal distribution retains memory of the full initial charge distribution, which is reflected in the projected ensemble (Chang et al., 2024).

Distinct universality classes arise:

Universality class	Dynamics/constraints	Limiting ensemble
Haar	No conservation	Haar-random on $\mathcal H_A$ (Mark et al., 2024)
Scrooge	Fixed $\rho_A$	Scrooge ensemble ($p(\psi)\propto \langle\psi|\rho_A|\psi\rangle$) (Mark et al., 2024, Liu et al., 2024)
Direct-sum Haar	Fixed charge sector	Direct sum over sectors, Haar within each (Chang et al., 2024)
Generalized Scrooge (GSE)	Nontrivial $p(Q)$, $Q$-revealing measurement	Weighted sum of sector Scrooge ensembles (Chang et al., 2024)
dGGE	Free integrable systems	Deep generalized Gibbs ensemble (2207.13628)
Generalized Hilbert–Schmidt	Mixed-state input, incomplete measurement	Random mixed-states (density matrices) (Yu et al., 12 May 2025, Sherry et al., 18 Jul 2025)

In continuous-variable (Gaussian) systems, the universal ensemble is the "Gaussian Scrooge distribution" of coherent states with Gaussian-distributed displacements (Liu et al., 2024).

4. Deep Ergodicity Breaking and Phase Transitions

Deep thermalization can exhibit sharply-defined phase transitions, invisible to standard thermalization diagnostics. A primary example is the coherence-induced transition in random permutation dynamics: the projected ensemble transitions from a minimally entropic classical bit-string ensemble (zero coherence) to a maximally entropic Haar ensemble (maximal coherence) as the total (input plus measurement-induced) coherence crosses a well-defined threshold. In the mixed-basis model, the critical boundary is $\alpha_0 + \alpha_m = 1$, and in the tilted-basis model, $$H_2(\cos^2(\theta_0/2)) + H_2(\cos^2(\theta_m/2)) = \ln 2$$ (Liu et al., 21 Oct 2025). Crucially, the subsystem's reduced density matrix remains maximally mixed in both phases; only higher moments of the projected ensemble diagnose the transition.

Analogous behavior appears under kinetic constraints, weak ETH violations, or in models with residual symmetries, where deep thermalization may fail even though local observables are thermalized, exposing a hierarchy of ergodicity (Bhore et al., 2023).

5. Mixed-State Generalizations and the Limitations of Pure-State Frameworks

For mixed initial states, the structure of deep thermalization departs fundamentally from the pure-state scenario. The projected ensemble, now comprising mixed states, cannot realize the pure-state Haar or Scrooge ensemble—even an infinitesimal admixture of mixedness in the global state destroys higher-moment $k$-designs for $k\ge2$ (Sherry et al., 18 Jul 2025). The resolution is a new maximum entropy construction: one purifies the initial mixed state by introducing an auxiliary system, constructs the pure-state Scrooge ensemble on the enlarged system, and traces out the auxiliary degrees of freedom. The resulting "mixed-state deep thermal ensemble" explicitly depends on the full spectrum (entropy) of the initial state. This framework dynamically emerges in chaotic dynamics and coincides with generalized Hilbert–Schmidt ensembles in solvable dual-unitary circuits (Sherry et al., 18 Jul 2025, Yu et al., 12 May 2025).

Incomplete or lossy measurement of the bath can likewise be captured: the limiting mixed-state projected ensemble corresponds to well-known random density matrix ensembles, and the dynamics exhibits sharp transitions in quantum teleportation fidelity at critical measurement thresholds (Yu et al., 12 May 2025).

6. Computational Equilibration, Nonlocality, and Experimental Realization

Recent studies have established that computationally efficient quantum circuits can realize deep thermalization in polylogarithmic depth, with circuit designs that ensure indistinguishability (to computationally bounded observers) from Haar randomness, both globally and under partial measurements, even with area-law entanglement (Chakraborty et al., 18 Jul 2025). Holographic deep thermalization protocols offer hardware-efficient routes to Haar-random state sampling with drastically reduced ancilla overhead, using sequential “scramble–measure–reset” cycles and furnishing rigorous bounds on frame-potential convergence and security (decoupling from adversarial side information) (Zhang et al., 2024).

Deep thermalization is highly nonlocal: the rate and uniformity of projected ensemble formation depends on global properties such as system topology and boundary conditions, with analytically tractable differences in convergence rates (e.g., periodic vs. open boundaries exhibiting a factor-of-two difference in deep thermalization velocity) (Shrotriya et al., 2023).

Experimental platforms, including superconducting qubits, Rydberg arrays, and linear optics, are increasingly able to access projected ensembles by implementing (partial) measurements on system complements, reconstructing moments of the distribution, and directly observing deep thermalization or its breakdown, as well as related teleportation and information-scrambling phenomena (Yu et al., 12 May 2025, Zhang et al., 2024).

7. Broader Implications: Resource Theories, Information Scrambling, and Universality

Deep thermalization underpins a universal mechanism for the saturation of maximum entropy and information scrambling in quantum dynamics. The projected ensemble formalism provides a versatile metrology tool: for quantifying the resource-generating power (RGP) of quantum channels in multiple resource theories (coherence, entanglement, magic), with twirling identities relating ensemble moments to physically meaningful resource monotones (Varikuti et al., 10 Dec 2025). Subsystem-level deep thermalization ensures that not only average quantities but the full statistics of subsystem resources thermalize, obeying exponential convergence laws set by the underlying circuit and symmetry structure.

Information-theoretically, the maximum entropy property makes deep-thermalized ensembles optimal in hiding information, saturating bounds on subentropy and rendering state discrimination maximally difficult (Mark et al., 2024, Liu et al., 2024). In practical terms, deep thermalization appears as the uniquely efficient route to random circuit sampling, rigorous benchmarking, and quantum cryptographic state preparation with provable security against general adversaries (Chakraborty et al., 18 Jul 2025, Zhang et al., 2024).

In summary, deep thermalization encapsulates a hierarchy of universal statistical behaviors—controlled by the interplay of dynamics, symmetry, conservation laws, and measurement structure—that generalize and transcend conventional quantum thermalization. It provides a rigorous operational framework for diagnosing both the success and limits of quantum ergodicity, for classifying nonlocal order and resource generation, and for benchmarking the complexity of quantum dynamics in both theoretical models and experimental implementations.