Quantum Orlicz spaces in information geometry

TL;DR

This paper introduces a quantum version of Orlicz spaces using a new norm based on finite entropy states, extending classical information geometry to quantum settings.

Contribution

It develops a quantum framework for Orlicz spaces, including the Luxemburg norm and affine structures, inspired by classical nonparametric estimation theories.

Findings

Defined a new norm finite for states with finite entropy

Constructed quantum analogs of tangent and cotangent spaces

Established a Holder-Orlicz inequality in the quantum context

Abstract

A start is made to redefining the topology of the spaces of normal states (density operators) by a new norm which is finite only for states of finite entropy. It is shown that a symmetrized version of the free energy difference between states can be used as a quantum version of the cosh Young function used in the theory of Orlicz space. The results form a quantum version of the classical treatment of nonparametric estimation by information geometry, in the work of Pistone and Sempi. We succeed in constructing the Luxemburg norm, the tangent space carrying the (+1)-affine structure, and the cotangent space carrying the (-1) affine structure, and we demonstrate the Holder-Orlicz inequality.