A characterization of extended monotone metrics

Classical information geometry has emerged from the study of geometrical aspect of the statistical estimation. Cencov characterized the Fisher metric as a canonical metric on probability simplexes from a statistical point of view, and Campbell extended the characterization of the Fisher metric from probability simplexes to positive cone . In quantum information geometry, quantum states which are represented by positive Hermitian matrices with trace one are regarded as an extension of probability distributions. A quantum version of the Fisher metric is introduced, and is called a monotone metric. Petz characterized the monotone metrics on the space of all quantum states in terms of operator monotone functions. A purpose of the present paper is to extend a characterization of monotone metrics from the space of all states to the space of all positive Hermitian matrices on finite dimensional Hilbert space. This characterization corresponds quantum modification of Campbell’s work.