Non-monotone metric on the quantum parametric model

Abstract

In this paper, we study a family of quantum Fisher metrics based on a convex mixture of two well-known inner products, which covers the well-known symmetric logarithmic derivative, the right logarithmic derivative, and the left logarithmic derivative Fisher metrics. We then define a two-parameter family of quantum Fisher metrics, which is not necessarily monotone. We derive a necessary and sufficient condition for this metric to be monotone. As an application of our proposed metric, we show several characterizations of quantum statistical models for the D-invariant model, asymptotically classical model, and classical model. In our study, the commutation super-operator introduced by Holevo plays a key role. This operator enables us to characterize properties of the tangent spaces of the quantum statistical model and to associate it to the Holevo bound in a unified manner.