Fundamentals of Continuous Variables

Abstract

The fundamentals introduced in Chap. 3, based on the four postulates of Quantum Mechanics, are concerned with discrete variables, with operators and quantum measurements specified by enumerable sets. But Quantum Information makes use of both discrete and continuous variables. The extension of fundamentals to continuous variables is done starting from the quantum harmonic oscillator, which is the basis of the theory of the electromagnetic field, in which the electromagnetic radiation is represented as a combination of harmonic oscillators. In this context, position and momentum (canonical variables), and annihilator and the creation operator (bosonic variables) are the fundamental operators for the development of the theory of continuous variables. The environment is an infinite dimensional Hilbert space, but an alternative environment is given by the phase space, a two-dimensional real space where quantum states of infinite dimensions are simply represented (in the single mode) by two functions of two real variables, the Wigner and the characteristic functions. These functions allow for the introduction of Gaussian states and Gaussian transformations. In addition to offering an easy description in terms of Gaussian functions, Gaussian states and transformations are of great practical relevance and represent the main tool of Quantum Information processing, with applications to quantum computation, quantum cryptography, and quantum communications. Coherent states are notable examples of Gaussian states, but in this chapter several other examples of Gaussian states are also considered. The most interesting applications, in particular the ones based on the entanglement, are concerned with continuous variables in the multimode, where the Hilbert space is given by \(N\) replicas (tensor product) of the space of the single mode and the phase space becomes \(2N\)-dimensional. The extension to multimode continuous variables represents the main difficulty of the chapter.