Information Geometry and Statistical Inference D

Variance and Fisher information are ingredients of the Cram er-Rao inequality. Fisher information is regarded as a Riemannian metric on a quantum statistical manifold and we choose monotonicity under coarse graining as the fundamental property. The quadratic cost functions are in a dual relation with the Fisher information quantities and they reduce to the variance in the commuting case. The scalar curvature in a certain geometry might be interpreted as an uncertainty on a statistical manifold. Information geometry has a surprising application to the theory of geometric mean of matrices. 1. The Cram er-Rao Inequality The Cram er-Rao inequality belongs to the basics of estimation theory in mathematical statistics. Its quantum analog was discovered immediately after the foundation of mathematical quantum estimation theory in the 1960's, see the book [10] of Helstrom, or the book [11] of Holevo for a rigorous summary of the subject. Although both the classical Cram er-Rao inequality and its quantum analog are as trivial as the Schwarz inequality, the subject takes a lot of attention because it is located on the highly exciting boundary of statistics, information and quantum theory. As a starting point we give a very general form of the quantum Cram er-Rao inequality in the simple setting of nite dimensional quantum mechanics. For 2 ( "; ") R a statistical operator is given and the aim is to estimate the value of the parameter close to 0. Formally is an n n positive semide nite matrix of trace 1 which describes a mixed state of a quantum mechanical system and we assume that is smooth (in ). Assume that an estimation is performed by the measurement of a self-adjoint matrix A playing the role of an observable. A is called locally unbiased estimator if (1) @ @ Tr A =0 = 1 : This condition holds if A is an unbiased estimator for , that is (2) Tr A = ( 2 ( "; ")): Supported by the Hungarian grant OTKA T032662.