We generalize Lévy’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 \({\mathcal {H}}\) , \({\textrm{GAP}}(\rho )\) 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 \(\Vert \rho \Vert \) 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 \({\textrm{GAP}}(\rho )\) , 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.
