The semigroup structure of gaussian channels
Teiko Heinosaari, Alexander Semenovich Holevo, Michael M. Wolf · 2010
We investigate the semigroup structure of bosonic Gaussian quantum channels. Par-ticular focus lies on the sets of channels which are divisible, idempotent or Markovian(in the sense of either belonging to one-parameter semigroups or being infinitesimal di-visible). We show that the non-compactness of the set of Gaussian channels allows forremarkable differences when comparing the semigroup structure with that of finite di-mensional quantum channels. For instance, every irreversible Gaussian channel is shownto be divisible in spite of the existence of Gaussian channels which are not infinitesimaldivisible. A simpler and known consequence of non-compactness is the lack of generatorsfor certain reversible channels. Along the way we provide new representations for classesof Gaussian channels: as matrix semigroup, complex valued positive matrices or in termsof a simple form describing almost all one-parameter semigroups.