In quantum estimation theory, the Holevo bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Holevo bound is defined by a solution of a minimization problem, and in general, an explicit solution is not known. When the dimension of Hilbert space is 2 and the number of parameters is 2, an explicit form of the Holevo bound was given by Suzuki. In this paper, we focus on a logarithmic derivative that lies between the symmetric logarithmic derivative (SLD) and the right logarithmic derivative parameterized by β ∈ [0, 1] to obtain lower bounds of the weighted trace of covariance of an unbiased estimator. We introduce the maximum logarithmic derivative bound as the maximum of bounds with respect to β. We show that all monotone metrics induce lower bounds, and the maximum logarithmic derivative bound is the largest bound among them. We show that the maximum logarithmic derivative bound has explicit solution when the d dimensional model has d + 1 dimensional $\mathcal{D}$ invariant extension of the SLD tangent space. Furthermore, when d = 2, we show that the maximization problem to define the maximum logarithmic derivative bound is the Lagrangian duality of the minimization problem to define the Holevo bound and is the same as the Holevo bound. This explicit solution is a generalization of the solution for a two-dimensional Hilbert space given by Suzuki. We also give examples of families of quantum states to which our theory can be applied not only for two-dimensional Hilbert spaces.

中文翻译：

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量子态估计上的最大对数导数界作为 Holevo 界的对偶

在量子估计理论中，Holevo 界被称为无偏估计量协方差的加权迹的下界。Holevo 界由最小化问题的解定义，一般情况下，显式解是未知的。当希尔伯特空间的维数为 2，参数个数为 2 时，Suzuki 给出了 Holevo 界的显式形式。在本文中，我们关注位于对称对数导数 (SLD) 和由β ∈ [0, 1]参数化的右对数导数之间的对数导数，以获得无偏估计量的协方差加权迹的下界。我们引入最大对数导数界作为关于β的界的最大值. 我们表明所有单调度量都会产生下界，最大对数导数界是其中最大的界。我们证明当d维模型有d + 1 维时，最大对数导数界有显式解$\mathcal{D}$SLD 切空间的不变扩展。此外，当d = 2 时，我们证明定义最大对数导数界的最大化问题是定义 Holevo 界的最小化问题的拉格朗日对偶，与 Holevo 界相同。这个显式解是 Suzuki 给出的二维希尔伯特空间解的推广。我们还给出了量子态族的例子，我们的理论不仅可以应用于二维希尔伯特空间。