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Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles

We consider an isolated macroscopic quantum system in a pure state $ψ_t$ evolving unitarily in a separable Hilbert space $\mathcal{H}$ and take for granted that different macro states $ν$ correspond to mutually orthogonal subspaces $\mathcal{H}_ν\subset\mathcal{H}$. Let $P_ν$ be the projection to $\mathcal{H}_ν$. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most $ψ_0\in\mathcal{H}_μ$ are such that for most $t\geq 0$, $\|P_νψ_t\|^2$ is close to a $t$- and $ψ_0$-independent value $M_{μν}$ provided that $M_{μν}$ is not too small. Here, ``most'' refers to the uniform distribution on the sphere $\mathbb{S}(\mathcal{H}_μ)$. In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix $ρ$ on $\mathcal{H}$, $\mathrm{GAP}(ρ)$ is the most spread out distribution on $\mathbb{S}(\mathcal{H})$ with density matrix $ρ$. We show that also for $\mathrm{GAP}(ρ)$-most $ψ_0\in\mathcal{H}$ for most $t\geq 0$, $\|P_νψ_t\|^2$ is close to a fixed value $M_{ρP_ν}$ (which must not be too small). Moreover, we prove a generalization for certain operators $B$ instead of $P_ν$ and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.