Quantum information is a rapidly advancing area of interdisciplinary research. It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement. This article reviews the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field.
CONTENTS
I. Introduction 513 II. Continuous Variables in Quantum Optics 516 A. The quadratures of the quantized field 516 B. Phase-space representations 518 C. Gaussian states 519 D. Linear optics 519 E. Nonlinear optics 520 F. Polarization and spin representations 522 G. Necessity of phase reference 523 III. Continuous-Variable Entanglement 523 A. Bipartite entanglement 525 1. Pure states 525 2. Mixed states and inseparability criteria 526 B. Multipartite entanglement 529 1. Discrete variables 529 2. Genuine multipartite entanglement 530 3. Separability properties of Gaussian states 530 4. Generating entanglement 531 5. Measuring entanglement 533 C. Bound entanglement 534 D. Nonlocality 534 1. Traditional EPR-type approach 535 2. Phase-space approach 536 3. Pseudospin approach 536 E. Verifying entanglement experimentally 537 IV. Quantum Communication with Continuous Variables 538 A. Quantum teleportation 540 1. Teleportation protocol 541 2. Teleportation criteria 543 3. Entanglement swapping 546 B. Dense coding 546 1. Information: A measure 547 2. Mutual information 547 3. Classical communication 547 4. Classical communication via quantum states 547 5. Dense coding 548 C. Quantum error correction 550 D. Quantum cryptography 550 1. Entanglement-based versus prepare and measure 550 2. Early ideas and recent progress 551 3. Absolute theoretical security 552 4. Verifying experimental security 553 5. Quantum secret sharing 553 E. Entanglement distillation 554 F. Quantum memory 555 V. Quantum Cloning with Continuous Variables 555 A. Local universal cloning 555 1. Beyond no-cloning 555 2. Universal cloners 556 B. Local cloning of Gaussian states 557 1. Fidelity bounds for Gaussian cloners 557 2. An optical cloning circuit for coherent states 558 C. Telecloning 559 VI. Quantum Computation with Continuous Variables 560 A. Universal quantum computation 560 B. Extension of the Gottesman-Knill theorem 563 VII. Experiments with Continuous Quantum Variables 565 A. Generation of squeezed-state EPR entanglement 565 1. Broadband entanglement via optical parametric amplification 565 2. Kerr effect and linear interference 567 B. Generation of long-lived atomic entanglement 568 C. Generation of genuine multipartite entanglement 569 D. Quantum teleportation of coherent states 569