Canonical Typicality For Other Ensembles Than Micro-Canonical

We generalize L\'evy's lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $\rho$ on a separable Hilbert space $\mathcal{H}$, GAP$(\rho)$ is the most spread out probability measure on the unit sphere of $\mathcal{H}$ that has density matrix $\rho$ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $\|\rho\|$ of $\rho$ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for ``most'' pure states $\psi$ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $\psi$-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time-evolution, for ``most'' pure states $\psi$ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $\psi_t$ is very close to a $\psi$-independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for GAP$(\rho)$, provided the density matrix $\rho$ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.