The set of states on CCR(H), the CCR algebra of a separable Hilbert space H, 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 CCR(H) that are invariant under the action of all automorphisms induced in second quantization by unitaries of H. 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.