Physics Reports 369 (2002) 431–548 www.elsevier.com/locate/physrep Fundamentals of quantum information theory Michael Keyl TU-Braunschweig, Institute of Mathematical Physics, Mendelssohnstrae 3, D-38106 Braunschweig, Germany Received 3 June 2002 editor: J. Eichler Abstract In this paper we give a self-contained introduction to the conceptional and mathematical foundations of quantum information theory. In the ÿrst part we introduce the basic notions like entanglement, channels, teleportation, etc. and their mathematical description. The second part is focused on a presentation of the quantitative aspects of the theory. Topics discussed in this context include: entanglement measures, chan- nel capacities, relations between both, additivity and continuity properties and asymptotic rates of quantum operations. Finally, we give an overview on some recent developments and open questions. c 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.−a; 03.65.−w Contents 1. Introduction ........................................................................................ 433 1.1. What is quantum information? ................................................................... 434 1.2. Tasks of quantum information .................................................................... 436 1.3. Experimental realizations ........................................................................ 438 2. Basic concepts ...................................................................................... 439 2.1. Systems, states and e=ects ....................................................................... 439 2.1.1. Operator algebras ........................................................................ 440 2.1.2. Quantum mechanics ...................................................................... 441 2.1.3. Classical probability ...................................................................... 442 2.1.4. Observables ............................................................................. 443 2.2. Composite systems and entangled states ........................................................... 444 2.2.1. Tensor products ......................................................................... 444 2.2.2. Compound and hybrid systems ............................................................ 445 2.2.3. Correlations and entanglement ............................................................. 446 2.2.4. Bell inequalities ......................................................................... 447 E-mail address: [email protected] (M. Keyl). 0370-1573/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S 0370-1573(02)00266-1 432 M. Keyl / Physics Reports 369 (2002) 431–548 2.3. Channels ...................................................................................... 448 2.3.1. Completely positive maps ................................................................. 449 2.3.2. The Stinespring theorem .................................................................. 450 2.3.3. The duality lemma ....................................................................... 450 2.4. Separability criteria and positive maps ............................................................ 451 2.4.1. Positivity ............................................................................... 451 2.4.2. The partial transpose ..................................................................... 452 2.4.3. The reduction criterion ................................................................... 453 3. Basic examples ..................................................................................... 453 3.1. Entanglement .................................................................................. 454 3.1.1. Maximally entangled states ................................................................ 454 3.1.2. Werner states ........................................................................... 455 3.1.3. Isotropic states .......................................................................... 456 3.1.4. OO-invariant states ....................................................................... 457 3.1.5. PPT states .............................................................................. 459 3.1.6. Multipartite states ........................................................................ 460 3.2. Channels ...................................................................................... 461 3.2.1. Quantum channnels ...................................................................... 461 3.2.2. Channels under symmetry ................................................................. 463 3.2.3. Classical channels........................................................................ 464 3.2.4. Observables and preparations .............................................................. 464 3.2.5. Instruments and parameter-dependent operations ............................................. 465 3.2.6. LOCC and separable channels ............................................................. 467 3.3. Quantum mechanics in phase space ............................................................... 468 3.3.1. Weyl operators and the CCR .............................................................. 468 3.3.2. Gaussian states .......................................................................... 469 3.3.3. Entangled Gaussians ..................................................................... 470 3.3.4. Gaussian channels ....................................................................... 472 4. Basic tasks ......................................................................................... 473 4.1. Teleportation and dense coding ................................................................... 473 4.1.1. Impossible machines revisited: classical teleportation ......................................... 474 4.1.2. Entanglement enhanced teleportation ....................................................... 474 4.1.3. Dense coding ........................................................................... 476 4.2. Estimating and copying ......................................................................... 477 4.2.1. Quantum state estimation ................................................................. 477 4.2.2. Approximate cloning ..................................................................... 478 4.3. Distillation of entanglement ...................................................................... 479 4.3.1. Distillation of pairs of qubits .............................................................. 480 4.3.2. Distillation of isotropic states .............................................................. 481 4.3.3. Bound entangled states ................................................................... 481 4.4. Quantum error correction ........................................................................ 482 4.5. Quantum computing ............................................................................ 485 4.5.1. The network model of classical computing .................................................. 485 4.5.2. Computational complexity................................................................. 486 4.5.3. Reversible computing .................................................................... 487 4.5.4. The network model of a quantum computer ................................................. 487 4.5.5. Simons problem ......................................................................... 490 4.6. Quantum cryptography .......................................................................... 491 5. Entanglement measures .............................................................................. 493 5.1. General properties and deÿnitions ................................................................. 493 5.1.1. Axiomatics .............................................................................. 493 M. Keyl / Physics Reports 369 (2002) 431–548 433 5.1.2. Pure states .............................................................................. 495 5.1.3. Entanglement measures for mixed states .................................................... 497 5.2. Two qubits .................................................................................... 498 5.2.1. Pure states .............................................................................. 499 5.2.2. EOFfor Bell diagonal states .............................................................. 500 5.2.3. Wootters formula ........................................................................ 501 5.2.4. Relative entropy for Bell diagonal states .................................................... 502 5.3. Entanglement measures under symmetry ........................................................... 503 5.3.1. Entanglement of formation ................................................................ 503 5.3.2. Werner states ........................................................................... 503 5.3.3. Isotropic states .......................................................................... 505 5.3.4. OO-invariant states ....................................................................... 505 5.3.5. Relative entropy of entanglement .......................................................... 507 6. Channel capacity .................................................................................... 509 6.1. The general case ..............................................................................