Relationship between deep thermalization and eigenstate thermalization

Determine whether refined eigenstate thermalization hypotheses characterizing higher-order fluctuations of local observables imply deep thermalization, whether deep thermalization implies those refined eigenstate thermalization properties, and under what dynamical or eigenstate conditions conventional thermalization implies deep thermalization.

Background

The paper presents deep thermalization as a stronger notion than conventional thermalization because it characterizes statistical fluctuations of projected states conditioned on bath measurements, rather than only averaged local observables. It contrasts this framework with refinements of the eigenstate thermalization hypothesis that address higher-order fluctuations of local observables.

The authors identify as unresolved whether either framework entails the other, and more generally what properties of the dynamics or eigenstates allow conventional thermalization to guarantee deep thermalization. They also connect this issue to thermalization timescales and approximately conserved quantities, but the central unresolved problem is the relationship among these forms of thermalization.

References

While deep thermalization characterizes a different type of fluctuation (over bath states), does either notion imply the other? More generally, under what conditions---in terms of dynamics or of eigenstates---does conventional thermalization imply deep thermalization?

— Deep thermalization and Hilbert space ergodicity  (2609.30248 - Mark et al., 24 Sep 2026) in Discussion and outlook, subsection “Connection with thermalization”

These discoveries show robust exceptions to the maximum entropy principle, but what is the full landscape of such exceptions? And which physical principles, beyond quantum resource constraints, underpin them? Further, are there connections to the rich physics of measurement-induced phase transitions, where sharp changes in entanglement structures in post-measurement states are similarly found?

— Deep thermalization and Hilbert space ergodicity  (2609.30248 - Mark et al., 24 Sep 2026) in Discussion and outlook, subsection “Phase transitions and universality classes”