Undressed metastable states: prove long-time stability without local dressing

Determine whether the nonperturbatively long preservation of local correlation functions established for the dressed metastable state U|ψ0⟩ also holds for the original undressed metastable pure state |ψ0⟩. Concretely, prove that, for spatially local lattice Hamiltonians H satisfying the paper’s assumptions, the reduced density matrices of |ψ0⟩ on any fixed finite region remain close (in trace norm) to their initial values up to the same prethermal timescale guaranteed for the dressed state, and provide an explicit bound analogous to Proposition/Corollary 4 for |ψ0⟩ itself.

Background

The paper proves that metastable states exhibit prethermal behavior by constructing a quasi-local dressing U that locally diagonalizes the Hamiltonian around the state, and then showing that all local correlation functions in the dressed state U|ψ0⟩ remain nearly constant up to nonperturbatively long times.

The authors note that physical intuition suggests similar stability for the original undressed state |ψ0⟩, but only provide a rigorous guarantee for the dressed state, leaving open whether the same bound can be established directly for |ψ0⟩.