We generalize Lévy's Lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a more general class of measures, so-called GAP measures. For any given density matrix rho on a separable Hilbert space H, GAP(rho) is the most spread out probability measure on the unit sphere of H that has density matrix rho and thus forms a natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ||rho|| of rho is small. With the help of this result we generalize the well-known and important phenomena of ''canonical typicality'' and ''dynamical typicality'' from the uniform distribution to GAP measures. Moreover, with the help of upper bounds for the GAP-variance of <psi|A|psi>, where A is a bounded operator, we show a generalization of ''normal typicality'' to GAP measures. These typicality statements hold true whenever all eigenvalues of rho are small. Since certain GAP measures are quantum analogues of the canonical ensemble of classical mechanics, our results can be regarded as expressing a version of equivalence of ensembles. The talk is based on joint work with Stefan Teufel and Roderich Tumulka.
