Freedman’s Theorem for Unitarily Invariant States on the CCR Algebra

Abstract

AbstractThe set of states on $$\textrm{CCR}({{\mathcal {H}}})$$ CCR ( H ) , the CCR algebra of a separable Hilbert space $${{\mathcal {H}}}$$ 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 $$\textrm{CCR}({{\mathcal {H}}})$$ CCR ( H ) that are invariant under the action of all automorphisms induced in second quantization by unitaries of $${{\mathcal {H}}}$$ 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.