Absence of integrals of motion implies typical thermalization of product states

We establish rigorous operator-space conditions for thermalization of typical initial states drawn from low-entanglement ensembles under a given Hamiltonian or Floquet evolution. The state ensemble can be either random or finite-temperature sets of product states or matrix product states. We introduce an ensemble variance norm of operators whose decay characterizes typical thermalization, and find that the dynamics of this norm is intimately related to the operator growth. Based on this, we formulate thermalization conditions in terms of the absence of simple slow operators: integrals of motion (IOMs) or spectrum generating algebras (SGAs) of the Hamiltonian H or Floquet unitary U that are mainly composed of low-weight Pauli strings. We also present a scalable numerical algorithm to identify such simple slow operators for any given H or U. Our results offer a rigorous formulation of the widely-held belief that systems without IOMs thermalize. Notably, by including approximate IOMs and SGAs, our formalism allows to characterize thermalization in finite time without relying on assumptions about the spectral properties of the Hamiltonian.