Quantum Orlicz spaces in information geometry

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.