量子统计中Cramér-Rao-Bhattacharya不等式的哲学意义
On the philosophy of Cramér-Rao-Bhattacharya Inequalities in Quantum Statistics
摘要：
对于有限能级量子系统的任意参数态族，我们将Fisher映射空间联系起来，并引入Cramér-Rao-Bhattacharya张量和Fisher信息形式的自然概念。这使我们抽象的Cramér-Rao-Bhattacharya下界的协方差阵的任何有限数量的参数函数的无偏估计。包括一些说明性的例子。在各种模技术假设下，我们的方法可应用于无限能级量子系统以及Borel空间上的古典概率分布的参数族。
英文摘要：
To any parametric family of states of a finite level quantum system we associate a space of Fisher maps and introduce the natural notions of Cramér-Rao-Bhattacharya tensor and Fisher information form. This leads us to an abstract Cramér-Rao-Bhattacharya lower bound for the covariance matrix of any finite number of unbiased estimators of parameteric functions. A number of illustrative examples is included. Modulo technical assumptions of various kinds our methods can be applied to infinite level quantum systems as well as parametric families of classical probability distributions on Borel spaces.
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