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.

在本文中，我们研究了基于两个著名内积的凸混合的量子Fisher度量族，其中涵盖了著名的对称对数导数，右对数导数和左对数导数Fisher度量。然后，我们定义量子Fisher度量的两个参数族，该族不一定是单调的。我们得出使该度量单调的必要和充分条件。作为我们提出的量度的一种应用，我们展示了D统计量模型，渐近经典模型和经典模型的量子统计模型的几种特征。在我们的研究中，Holevo推出的换向超级算子起着关键作用。