Freedman's theorem for unitarily invariant states on the CCR algebra

Abstract

The set of states on $\mathrm{CCR}(\mathrm{ch})$, the CCR algebra of a separable Hilbert space $\mathrm{ch}$, is here looked at as a natural object to obtain a non-commutative version of Freedman's theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of $\mathrm{CCR}(\mathrm{ch})$ that are invariant under the action of all automorphisms induced in second quantization by unitaries of $\mathrm{ch}$. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least $1$.

22 pages