Finite de Finetti theorem for infinite-dimensional systems

We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state vertical bar Psi ><Psi vertical bar chosen from a family of subsets {C-n} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {C-n}.