Quantum Orlicz Spaces in Information Geometry

Springer Science and Business Media LLC - 2004
R. F. Streater1
1Department of Mathematics, King's College of London, Strand, WC2R 2LS, UK

Tóm tắt

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 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 ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − 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.

Từ khóa


Tài liệu tham khảo

10.1007/978-1-4612-5056-2

Amari S.-I., 1993, Methods of Information Geometry

Ekeland I., 1976, Convex Analysis and Variational Problems

10.1142/S0219025799000096

Hiriart-Urruty J.-P., 1993, Convex Analysis and Minimisation Algorithms I

10.1016/S0034-4877(00)90003-X

Krasnoselski M. A., 1961, Convex Functions and Orlicz Spaces

10.1002/mana.19901470114

10.1063/1.533053

10.1016/0001-8708(73)90011-X

Musielak J., 1980, Orlicz Spaces and Modular Spaces, 1034

10.1016/0024-3795(94)00211-8

10.1088/0305-4470/35/4/305

10.1007/978-3-642-87665-3

10.1214/aos/1176324311

Rao M. M., 1992, Theory of Orlicz Spaces

Streater R. F., Proc. Steklov Institute of Mathematics, 228, 205

Streater R. F., Conference Proc., Canadian Math. Soc., 29, 603

10.1023/B:OPSY.0000024757.25401.db

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Thông tin tác giả