Quantum Orlicz Spaces in Information Geometry

Let H0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let Xǫ denote the set of ǫ-bounded forms, i.e ‖(H0+C) X(H 0 +C)‖ < C for some C > 0. Let X := Span ∪ǫ∈(0,1/2] Xǫ. Let M denote the underlying set of the quantum information manifold of states of the form ρ X = e0X , X ∈ X . We show that if Tr e0 = 1, 1. the map Φ, Φ(X) = 1 2 Tr ( e0 + e0 ) − 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 Cρ 0 ≤ σ ≤ Cρ 0 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.