Statistical geometry in quantum mechanics

A statistical model M is a family of probability distributions, characterized by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square-integrable functions. More precisely, by consideration of the square-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 principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on variance lower bounds for statistical estimation, and establish generalizations of the classical Cramer-Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration 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 compatible, complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert-space geometry. The application of generalized variance bounds in the case of quantum statistical estimation leads to a set of higher-order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.