Open Systems & Information Dynamics

Quantum Orlicz Spaces in Information Geometry

R. F. Streater

Published Dec 1, 2004• 52 citations

Abstract

Let H 0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let χ ɛ denote the set of ɛ-bounded forms, i.e., ∥(H 0 +C) −1/2−ɛ X(H 0 +C) −1/2+ɛ ∥ < C for some C > 0. Let χ := Span ∪ ɛ∈(0,1/2] χ ɛ . Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρ x = e −H 0 −X−ψx , X ∈ χ. We show that if Tr e −H 0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

DOI: 10.1007/s11080-004-6626-2

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