Canonical Typicality for Other Ensembles than Micro-canonical

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 ρ on a separable Hilbert space H, GAP(ρ) is the most spread-out probability measure on the unit sphere of H that has density matrix ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ‖ρ‖ of ρ 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 ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a ψ-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state ψt is very close to a ψ-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(ρ), provided the density matrix ρ 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.