Purity and local entropy in product hilbert space

Abstract

A property might be called "typical" for some ensemble of items, if this property will show up with high probability for any randomly selected ensemble member (cf. Chap. 6). Based on the (unitarily invariant) distribution function of pure states within a given Hilbert space of finite dimension ntot, we look for properties of such pure states exhibiting such a type of typicality. As a pertinent example we will show here that for a bipartite Hilbert space of dimension ntot = ng · nc the maximum local entropy (minimum local purity) within subsystem g (dimension ng) becomes typical, provided ng ≪ nc. Note that this observation is a consequence of the tensor space and virtually independent of the respective physical system. © 2009 Springer-Verlag Berlin Heidelberg.