Continuous Quantum Systems. I

Ola Bratteli, Derek W. Robinson

Mauro Paternostro

This paper presents the basic features of quantum simulation based on continuous-variable systems. Far from attempting at being complete, the paper is a compendium of the limitations, possibilities, and perspectives in quantum simulations with such a class of systems. It includes a short summary on the phase-space description of basic continuous-variable states and operations and a gentle introduction to the most crucial results on the simulation of Hamiltonians for continuous-variable systems.

Pierre Vernaz-Gris, Andreas Ketterer, Arne Keller +3 more

Observables with a continuous spectrum are known to fundamentally differ from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. Here we show that it is possible to manipulate, detect, and classify continuous-variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define, and adapt, to arbitrary continuous quantum systems, quantum protocols, and algorithms developed for discrete systems.

Takaya Matsuura

Carlo Presilla A, Ubaldo Tambini

John R. Klauder

The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.