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Quantum Information with Continuous Variables of Atoms and Light

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December 20, 2006 12:11 WSPC/Trim Size: 9in x 6in for Review Volume cerf˙book This page intentionally left blank

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Quantum Information with Continuous Variables of Atoms and Light Editors N. J. CERF Université Libre d e Bruxelles, Belgium G. LEUCHS Universität Erlangen -Nürnberg, Germany E. S. POLZIK Niels Bohr Institute, Denmark Imperial College Press ICP

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Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. QUANTUM INFORMATION WITH CONTINUOUS VARIABLES OF ATOMS AND LIGHT Copyright © 2007 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-86094-760-5 ISBN-10 1-86094-760-3 ISBN-13 978-1-86094-776-6 (pbk) ISBN-10 1-86094-776-X (pbk) Printed in Singapore. Magdalene - Quantum Information.pmd 1 12/21/2006, 2:55 PM

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December 21, 2006 11:32 WSPC/Trim Size: 9in x 6in for Review Volume cerf_book . . . continuous quantum variables are the language used in the original formulation of the EPR gedankenexperiment: Thus, by measuring either P [the momentum] or Q [the coordinate of the �rst system] we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P [: : :] or the value of the quantity Q [: : :]. In accordance with our criterion of reality, in the �rst case we must consider the quantity P as being an element of reality, in the second case the quantity Q is an element of real- ity. But, as we have seen, both wave functions [the eigenfunctions of P and Q] belong to the same reality. A. Einstein, B. Podolsky and N. Rosen (1935)

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December 20, 2006 12:11 WSPC/Trim Size: 9in x 6in for Review Volume cerf˙book Preface This book is a joint eﬀort of a number of leading research groups actively developing the ﬁeld of quantum information processing and communication (QIPC) with continuous variables. The term “continuous” refers to the fact that the description of quantum states within this approach is carried out in the phase space of canonical variables, x and p, which are indeed continuous variables over an inﬁnite dimensional Hilbert space. Historically, the ﬁeld of QIPC with continuous variables has dealt mostly with Gaussian states, such as coherent states, squeezed states, or Einstein-Podolsky-Rosen (EPR) two-mode entangled states. A powerful mathematical formalism for Gaussian states, which are completely described by only ﬁrst and second order momenta, is presented in the ﬁrst part of this book in the chapters by G. Adesso and F. Illuminati (entanglement properties of Gaussian states) and by J. Eisert and M. M. Wolf (Gaussian quantum channels). This is a useful tool in the study of entanglement properties of harmonic chains (see chapter by K. M. R. Audenaert et al.), as well as in the description of quantum key distribution based on coherent states (see chapter by F. Grosshans et al.). A more exotic topic involving Gaussian states is covered in the chapter by O. Kru¨ger and R. F. Werner (Gaussian quantum cellular automata). Gaussian operations on Gaussian states alone do not allow for the puriﬁ- cation and distillation of continuous-variable entanglement, features which are critical for error corrections in QIPC, so that the recourse to non- Gaussian operations is necessary (see chapter by J. Fiura´ˇsek et al.). Non- Gaussian operations are also crucial in order to build loophole-free Bell tests that rely on homodyne detection (see chapter by R. Garc´ıa-Patr´on). Interestingly, the continuous-variable formalism is also appropriate for the analysis of non-Gaussian states, such as Fock states, qubit (quantum bit) states, and coherent superposition (Schr¨odinger cat) states. Indeed, the Wigner function over an inﬁnite dimensional Hilbert space provides the most complete description of any state, including a discrete variable, qubit state. The Hilbert space may be spanned by the Fock state basis in the case of a single ﬁeld mode, or, in the case of single photons, by the spectral mode vii

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December 20, 2006 12:11 WSPC/Trim Size: 9in x 6in for Review Volume cerf˙book viii Preface functions. The characterization of such non-Gaussian states by homodyne tomography is reviewed in the chapter by G. M. D’Ariano et al. Then, re- cent theoretical developments in the generation of particular non-Gaussian states (Schro¨dinger cat states) are presented in the chapter by H. Jeong and T. C. Ralph. Continuous variables have played a particularly important role in QIPC with light, due to the highly eﬃcient and well experimentally developed method of “homodyne detection”, which provides a direct access to the canonical variables of light. This area of “optical continuous variables” is covered in the second part of this book. Here, the variables x and p are the two quadrature phase operators associated with the sine and cosine components of the electromagnetic ﬁeld. By mixing the quantum light ﬁeld under investigation with a strong classical “local oscillator” light on a beam splitter, the variables x and p can readily be observed, and hence a com- plete description of the quantum ﬁeld is obtained. If one takes into account the polarization of light as an additional degree of freedom, the Stokes operators have to be introduced and the notions of polarization squeez- ing and polarization entanglement arise, as described in the chapter by N. Korolkova. Several recent experiments with continuous variables of light are pre- sented in this part of the book. For example, the chapters by J. Laurat et al., O. Glo¨ckl et al., and V. Josse et al. present the generation of EPR en- tangled light via the optical nonlinearities provided by solid state materials and cold atoms. Some other chapters present several applications of opti- cal continuous variables to QIPC protocols, such as quantum teleportation by N. Takei et al., quantum state sharing by T. Tyc et al., and quantum cloning by U. L. Andersen et al. Applications of continuous-variable squeez- ing to ultra-precise measurements are covered in the chapters by C. Fabre et al. (quantum imaging) and by R. Schnabel (towards squeezing-enhanced gravitational wave interferometers). For single-photon states, the concept of canonical continuous variables can be transferred to other observables, e.g. the position x and wave vector k, as shown in the chapter by L. Zhang et al. The non-Gaussian operations such as photon counting combined with the continuous-variable homodyne-based analysis of the light conditioned on photon counting take QIPC with optical continuous variables into a new domain. This domain, where the puriﬁcation of entanglement and er- ror correction is, in principle, possible, is explored experimentally in the chapters by J. Wenger et al. (photon subtracted squeezed states) and by

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December 20, 2006 12:11 WSPC/Trim Size: 9in x 6in for Review Volume cerf˙book Preface ix A. I. Lvovsky and M. G. Raymer (single-photon Fock states). The latter chapter reports on the progress in experimental quantum tomography and state reconstruction. Another avenue in QIPC with continuous variables has opened up when it was realized that multi-atomic ensembles can well serve as eﬃcient stor- age and processing units for quantum information. The third part of this book is devoted to the development and application of this approach based on “atomic continuous variables”. The quantum interface between light pulses carrying quantum information and atomic processors has become an important ingredient in QIPC, as some of the most spectacular recent de- velopments of the light-atoms quantum interface have been achieved with atomic ensembles. The continuous-variable approach to atomic states has proven to be very competitive compared to the historically ﬁrst single atom and cavity QED approach. The theory of quantum non-demolition measurement on light transmit- ted through atoms, quantum feedback, and multi-pass interaction of light with atoms, is presented in the chapters by L. B. Madsen and K. Mølmer and by R. van Handel et al. Experiments on spin squeezing of atoms are de- scribed in the chapter by J. M. Geremia, while the theory and experiments of EPR entanglement of distant atomic objects and quantum memory for light are presented in the chapter by K. Hammerer et al. Atomic ensembles can also serve as sources of qubit-type entanglement. In this case, a single qubit state is distributed over the entire multi-atomic ensemble, providing thus a conceptual bridge between a discrete computational variable and a continuous (or collective) variable used as its physical implementation. The work towards the implementation of a promising proposal for the generation of such type of entanglement conditioned on photon detection (the Duan- Lukin-Cirac-Zoller protocol) is presented in the chapter by C. W. Chou et al. Interestingly, such an analysis of qubits in the continuous-variable language makes the old sharp boundary between continuous and discrete variables softer. Finally, the theory of decoherence suppression in quantum memories for photons is discussed in the chapter by M. Fleischhauer and C. Mewes. In summary, this book is aimed at providing a comprehensive review of the main recent progresses in continuous-variable quantum information processing and communication, a ﬁeld which has been rapidly develop- ing both theoretically and experimentally over the last ﬁve years. It was originally intended to review the main advances that had resulted from the project “Quantum Information with Continuous Variables” (QUICOV)

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December 20, 2006 12:11 WSPC/Trim Size: 9in x 6in for Review Volume cerf˙book x Preface funded by the European Commission from 2000 to 2003. However, given the unexpected pace at which new paradigms and applications continued to appear, it soon became clear that this objective had become too restric- tive. Instead, this book evolved into a compilation of the even more re- cent achievements that were reported in the series of workshops especially devoted to continuous-variable QIPC that took place in Brussels (2002), Aix-en-Provence (2003), Veilbronn (2004), and Prague (2005). Yet, the pic- ture would not have been complete without the contributions of several additional world experts, which have rendered this book fairly exhaustive. We are conﬁdent that the various directions explored in the 27 chapters of this book will form a useful basis in order to approach continuous-variable QIPC. This is, however, probably not the end of the story, and we expect that future developments in this ﬁeld will open new horizons in quantum state engineering, quantum computing and communication. We warmly thank Gerlinde Gardavsky for her careful work on preparing the lay-out, correcting and proof-reading this book. Nicolas J. Cerf Gerd Leuchs Eugene S. Polzik Editors

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