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Quantum Orlicz Spaces in Information Geometry

Authors:

R F Streater

Affiliation:

(1) Department of Mathematics, King’s College of London, Strand, WC2R 2LS, UK

Abstract:

Let H₀ be a selfadjoint operator such that Tr $e^{\beta H_0}$ is of trace class for some $\beta < 1$ , and let $\chi_\epsilon$ denote the set of ε-bounded forms, i.e., $||(H_0+C)^{-1/2-\epsilon}X(H_0+C)^{-1/2+\epsilon}|| < C$ for some $$ C > 0 $$

. Let χ := Span $\cup_{\epsilon \in (0,1/2]}\chi_\epsilon$ . Let ${\cal M}$ denote the underlying set of the quantum information manifold of states of the form $\rho_X = e{-H_0-X-\phi_X}, X \in \chi$ . We show that if Tr $e^{-H_0} = 1$ ,

the map Φ,

$\Phi(X)= \frac{1}{2}{\rm Tr}\left(e^{-H_0+x} + e^{-H_0-x}\right) -1$

is a quantum Young function defined on χ

The Orlicz space defined by Φ is the tangent space of ${\cal M}$ at ρ₀; its affine structure is defined by the (+1)-connection of Amari

The subset of a ‘hood of ρ₀, consisting of p-nearby states (those $\sigma \in {\cal M}$ obeying $C^{-1}\rho^{1+p} \leq \sigma \leq C\rho^{1-p}$ 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 ${\cal M}$

These dual structures extend to the completions in the Luxemburg norms.

Presented at the 36th Symposium on Mathematical Physics, ‘Open Systems & Quantum Information’, Toruń, Poland, June 9-12, 2004.

Keywords:

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