Quantum Orlicz Spaces in Information Geometry

摘要

Let Ho be a selfadjoint operator such that Tre~(-βH_0) is of trace class for some β < 1, and let X_ε denote the set of ε-bounded forms, i.e., ‖(H_0+C)~(-1/2-ε)X(H_0+C)~(-1/2+ε)‖ < C for some C > 0. Let X := Span ∪_(ε∈(0,1/2]) Χ_ε. Let M denote the underlying set of the quantum information manifold of states of the form ρ_x = e~(-H_0-X-ΨX), X ∈ X. We show that if Tr e~(H_0) = 1, 1. the map Φ, Φ(X) = 1/2Tr (e~(-H_0+X) +e~(-H_0-X) - 1 is a quantum Young function defined on X 2. The Orlicz space defined by Φ is the tangent space of M at ρ_0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a 'hood of ρ_0, consisting of p-nearby states (those σ ∈ M obeying C~(-1)ρ~(1+p) ≤ σ ≤ Cρ~(1-p) for some C > 1) admits a flat affine connection known as the (-1) connection, and the span of this set is part of the cotangent space of M 4. These dual structures extend to the completions in the Luxemburg norms.