Geometry and Probability In Quantum Theory

It is well known now that many aspects of quantum theory admit a natural interpretation in the classical geometry of complex projective space. It is curious that this approach to the theory seems to have been overlooked by the founders of the subject. Dirac, to be sure, was well aware of the advantages of projective geometry, and evidently carried out many of his private calculations in that way; but it seems that he was unaware of the natural metrical and symplectic geometries associated with the space of pure states in quantum mechanics.

We know now that the pure state space, when it is regarded as complex projective space endowed with the Fubini-Study metric, has a Riemannian structure, and that much of the standard theory can be developed in the language of this geometry. For example, the transition probability between two states can be expressed as a function of the geodesic distance between the two states in this geometry. For a long time the geometric approach was to some extent neglected, perhaps because its development in the infinite dimensional situation of a general Hilbert space involves technical issues. But over the last decade or two, with the advent of quantum information and various allied lines of investigation that have tended to emphasize the role of finite-dimensional state spaces, the situation has changed, and the geometric method in this context provides a powerful tool, allowing one to bring into play a variety of different constructions from classical algebraic geometry as an aid to understanding quantum theory and gaining new insights.

In this talk I present an overview of such methods, focusing on probabilistic aspects of the theory, especially the role of general measurements (POVMs) and various explicit geometrical constructions associated with these operations (including for example  the construction of SIC POVMs).

This work builds on material developed largely with D.C. Brody and other collaborators --- for background reading see, for example, D.C. Brody & L.P. Hughston (2000) Geometric Quantum Mechanics, J. Geometry and Physics 38, 19-53, and references cited therein.