A Primer on Hilbert Space Theory

R. D. Hazeltine

We introduce the key ideas and mathematical tools central to modern discussions of collapse. We assume knowledge of basic quantum mechanics -- the Schrödinger evolution of states in Hilbert space, the Born probability rule, the Dirac formalism, and so on -- while explicating such ideas as projectors, density operators, Bell inequalities, entanglement and the Lindblad equation. In other words this work is a \emph{primer} in the simplest sense of that word: without taking a stand on the major questions, it is intended to make an enormous and growing body of research more accessible.

Paul Busch, PEKKA J. LAHTI, Juha-Pekka Pellonpää +1 more

Pankaj Sharan

Alessio Serafini

Quantum Continuous Variables introduces the theory of continuous variable quantum systems, from its foundations based on the framework of Gaussian states to modern developments, including its applications to quantum information and forthcoming quantum technologies. This new book addresses the theory of Gaussian states, operations, and dynamics in great depth and breadth, through a novel approach that embraces both the Hilbert space and phase descriptions. The volume includes coverage of entanglement theory and quantum information protocols, and their connection with relevant experimental set-ups. General techniques for non-Gaussian manipulations also emerge as the treatment unfolds, and are demonstrated with specific case studies. This book will be of interest to graduate students looking to familiarise themselves with the field, in addition to experienced researchers eager to enhance their understanding of its theoretical methods. It will also appeal to experimentalists searching for a rigorous but accessible treatment of the theory in the area.

Songbo Xie

Stephen Bruce Sontz

In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.