Quantum Cloning with Continuous Variables

Quantum information theory has developed dramatically over the past decade, driven by the prospects of quantum-enhanced communication and computation systems. Among the most striking successes, one finds for example the discovery of quantum factoring, quantum key distribution, or quantum teleportation. Most of these concepts were initially developed for discrete quantum variables, in particular quantum bits, which have now become the symbol of quantum information. Recently, however, a lot of attention has been devoted to investigating the use of continuous-variable systems in quantum informational or computational processes. Continuous-spectrum quantum variables, for example the quadrature components of a light mode, may be easier to manipulate than quantum bits. It is actually sufficient to process squeezed states of light into linear optics circuits in order to perform various quantum information processes over continuous variables [1]. As reported in the present book, variables with a continuous spectrum have been shown to be useful to carry out quantum teleportation, quantum entanglement purification, quantum error correction, or even quantum computation. In this Chapter, the issue of cloning a continuous-variable quantum system will be analyzed, and a Gaussian cloning transformation will be introduced. Cloning machines, that is, transformations that achieve the best approximate copying of a quantum state compatible with the no-cloning theorem, have been a fundamental research topic over the last five years (see e.g. [2] for an overview). This question is of particular significance given the close connection between