The quantum information manifold for ε-bounded forms

Let H0 ≥ I be a self-adjoint operator and let V be a form-small perturbation such that , where ϵ ϵ (0, 12) and R0 = H−10. Suppose that there exists a positive β < 1 such that Z0 ≔ Tr e−βH0 < ∞. Let Z ≔ Tr e−(H0+V). Then we show that the free energy Ψ = log Z is an analytic function of V in the sense of Fréchet, and that the family of density operators defined in this way is an analytic manifold.