Characterization of symmetric monotone metrics on the state space of quantum systems

TL;DR

This paper fully characterizes symmetric monotone metrics on quantum state spaces, providing explicit formulas and a continuous spectrum connecting the extremal metrics, advancing understanding of quantum Fisher information geometry.

Contribution

It offers a complete, explicit characterization of Morozova-Chentsov functions for symmetric monotone metrics, including a continuous bridge between extremal metrics.

Findings

Derived a closed-form formula for Morozova-Chentsov functions.

Established a continuous connection between the smallest and largest metrics.

Enhanced understanding of quantum Fisher information geometry.

Abstract

The quantum Fisher information is a Riemannian metric, defined on the state space of a quantum system, which is symmetric and decreasing under stochastic mappings. Contrary to the classical case such a metric is not unique. We complete the characterization, initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed and tractable formula for the set of Morozova-Chentsov functions. In addition, we provide a continuously increasing bridge between the smallest and largest symmetric monotone metrics.