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

We consider an isolated macroscopic quantum system in a pure state $\psi_t$ evolving unitarily in a separable Hilbert space $\mathcal{H}$ and take for granted that different macro states $\nu$ correspond to mutually orthogonal subspaces $\mathcal{H}_\nu\subset\mathcal{H}$. Let $P_\nu$ be the projection to $\mathcal{H}_\nu$. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most $\psi_0\in\mathcal{H}_\mu$ are such that for most $t\geq 0$, $\|P_\nu\psi_t\|^2$ is close to a $t$- and $\psi_0$-independent value $M_{\mu\nu}$ provided that $M_{\mu\nu}$ is not too small. Here, ``most'' refers to the uniform distribution on the sphere $\mathbb{S}(\mathcal{H}_\mu)$. 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 $\rho$ on $\mathcal{H}$, $\mathrm{GAP}(\rho)$ is the most spread out distribution on $\mathbb{S}(\mathcal{H})$ with density matrix $\rho$. We show that also for $\mathrm{GAP}(\rho)$-most $\psi_0\in\mathcal{H}$ for most $t\geq 0$, $\|P_\nu\psi_t\|^2$ is close to a fixed value $M_{\rho P_\nu}$ (which must not be too small). Moreover, we prove a generalization for certain operators $B$ instead of $P_\nu$ 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.