STATISTICAL GEOMETRY IN QUANTUM-MECHANICS

A statistical model M is a family of probability distributions, charac terized by a set of continuous parameters known as the parameter space . This possesses natural geometrical properties induced by the embeddi ng of the family of probability distributions into the space of all sq uare-integrable functions. More precisely, by consideration of the squ are-root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the 'state space' of the probability distributions, and the geometry of the given statistical model, can be described in terms of the embedding of M in S. The geometry in question is characterized by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the prin ciples of classical statistical inference in a natural geometric setti ng. In particular, we focus attention on variance lower bounds for sta tistical estimation, and establish generalizations of the classical Cr amer-Rao and Bhattacharyya inequalities, described in terms of the geo metry of the underlying real Hilbert space. As a comprehensive illustr ation of the utility of the geometric framework, the statistical model M is then specialized to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a comp atible, complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterprete d in the language of real Hilbert-space geometry. The application of g eneralized variance bounds in the case of quantum statistical estimati on leads to a set of higher-order corrections to the Heisenberg uncert ainty relations for canonically conjugate observables.