Statistical Geometry in Quantum Mechanics

A statistical model M is a family of probability distributions, characterised 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 characterised 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 the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér–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 specialised 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 generalised 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.