On the philosophy of cramér-rao-bhattacharya inequalities in quantum statistics to my revered guru c. R. rao for revealing the mystries of chance

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Research Article

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Sankhya: The Indian Journal of Statistics


To any parametric family of states of a finite level quantum system we asso-ciate 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 are 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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