Concentration of measure for thermal distributions of quantum states

Mathphys Analysis Seminar

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 the 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 phenomenon of ''canonical typicality'' to GAP measures. 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. So far, canonical typicality is known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble. Our result shows that canonical typicality holds in general for systems described by a density matrix with 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. The talk is based on joint work with Stefan Teufel and Roderich Tumulka.