Some operator monotone functions

TL;DR

This paper characterizes a class of operator monotone functions and applies these findings to develop new quantum Fisher information metrics for quantum state spaces.

Contribution

It introduces a new family of operator monotone functions and derives associated canonical representations, enabling the construction of novel quantum monotone metrics.

Findings

Proved operator monotonicity of specific functions for 0 < p < q < 1

Derived canonical representation formulas for these functions

Developed new quantum Fisher information metrics based on these functions

Abstract

We prove that the functions t -> (t^q-1)(t^p-1)^{-1} are operator monotone in the positive half-axis for 0 < p < q < 1, and we calculate the two associated canonical representation formulae. The result is used to find new monotone metrics (quantum Fisher information) on the state space of quantum systems.