Covariance and Fisher information in quantum mechanics

Variance and Fisher information are ingredients of the Cramér-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts. We emphasis that Fisher informations are obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold. On the one hand standard quantum mechanics is a statistical theory, on the other hand, there is a so-called geometrical approach to mathematical statistics [1, 4]. In this paper the two topics are combined and the concept of covariance and Fisher information is studied from an abstract poit of view. We start with the Cramér-Rao inequality to realize that the two concepts are very strongly related. What they have in common is a kind of monotonicity property under coarse grainings. (Formally the monotonicity of covariance is a bit difference from that of Fisher information.) Monotone quantities of Fisher information type determine a superoperator J which gives immediately a kind of generalized covariance. In this way a one-to-one correspondence is established between the candidates of the two concepts. In the paper we prove a Cramér-Rao type inequality in the setting of generalized variance and Fisher information. Moreover, we argue that the scalar curvature of the Fisher information Riemannian metric has a statistical interpretation. This gives interpretation of an earlier formulated but still open conjecture on the monotonicity of the scalar curvature. 1 The Cramér-Rao inequality for an introduction The Cramér-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 [13] of Helstrom, or the book [14] of Holevo for a rigorous summary of the subject. Although both the