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Quantum Systems, Channels, Information A Mathematical Introduction

:	Alexander S. Holevo
:	Quantum Systems, Channels, Information A Mathematical Introduction
:	Walter de Gruyter GmbH& Co.KG
:	9783110273403
:	De Gruyter Studies in Mathematical PhysicsISSN
:	1
:	CHF 136.40
:

:	Theoretische Physik
:	English

:	362
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

< >The subject of this book is theory of quantum system presented from information science perspective. The central role is played by the concept of quantum channel and its entropic and information characteristics. Quantum information theory gives a key to understanding elusive phenomena of quantum world and provides a background for development of experimental techniques that enable measuring and manipulation of individual quantum systems. This is important for the new efficient applications such as quantum computing, communication and cryptography. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. This book gives an accessible, albeit mathematically rigorous and self-contained introduction to quantum information theory, starting from primary structures and leading to fundamental results and to exiting open problems.

< >Alexander S. Holevo, Steklov Mathematical Institute, Moscow, Russia.

	Preface	5
	I Basic structures	15
	1 Vectors and operators	17
	1.1 Hilbert space	17
	1.2 Operators	18
	1.3 Positivity	19
	1.4 Trace and duality	20
	1.5 Convexity	22
	1.6 Notes and references	23
	2 States, observables, statistics	24
	2.1 Structure of statistical theories	24
	2.1.1 Classical systems	24
	2.1.2 Axioms of statistical description	25
	2.2 Quantum states	28
	2.3 Quantum observables	30
	2.3.1 Quantum observables from the axioms	30
	2.3.2 Compatibility and complementarity	32
	2.3.3 The uncertainty relation	35
	2.3.4 Convex structure of observables	36
	2.4 Statistical discrimination between quantum states	39
	2.4.1 Formulation of the problem	39
	2.4.2 Optimal observables	39
	2.5 Notes and references	45
	3 Composite systems and entanglement	48
	3.1 Composite systems	48
	3.1.1 Tensor products	48
	3.1.2 Naimark’s dilation	50
	3.1.3 Schmidt decomposition and purification	52
	3.2 Quantum entanglement vs “local realism”	55
	3.2.1 Paradox of Einstein-Podolski-Rosen and Bell’s inequalities	55
	3.2.2 Mermin-Peres game	59
	3.3 Quantum systems as information carriers	61
	3.3.1 Transmission of classical information	61
	3.3.2 Entanglement and local operations	62
	3.3.3 Superdense coding	63
	3.3.4 Quantum teleportation	64
	3.4 Notes and references	66
	II The primary coding theorems	69
	4 Classical entropy and information	71
	4.1 Entropy of a random variable and data compression	71
	4.2 Conditional entropy and the Shannon information	73
	4.3 The Shannon capacity of the classical noisy channel	76
	4.4 The channel coding theorem	78
	4.5 Wiretap channel	83
	4.6 Gaussian channel	85
	4.7 Notes and references	86
	5 The classical-quantum channel	88
	5.1 Codes and achievable rates	88
	5.2 Formulation of the coding theorem	89
	5.3 The upper bound	92
	5.4 Proof of the weak converse	97
	5.5 Typical projectors	101
	5.6 Proof of the Direct Coding Theorem	106
	5.7 The reliability function for pure-state channel	109
	5.8 Notes and references	112
	III Channels and entropies	115
	6 Quantum evolutions and channels	117
	6.1 Quantum evolutions	117
	6.2 Completely positive maps	120
	6.3 Definition of the channel	126
	6.4 Entanglement-breaking and PPT channels	128
	6.5 Quantum measurement processes	131
	6.6 Complementary channels	133
	6.7 Covariant channels	138
	6.8 Qubit channels	141
	6.9 Notes and references	143
	7 Quantum entropy and information quantities	146
	7.1 Quantum relative entropy	146
	7.2 Monotonicity of the relative entropy	147
	7.3 Strong subadditivity of the quantum entropy	152
	7.4 Continuity properties	154
	7.5 Information correlation, entanglement of formation and conditional entropy	156
	7.6 Entropy exchange	161
	7.7 Quantum mutual information	163
	7.8 Notes and references	165
	IV Basic channel capacities	167
	8 The classical capacity of quantum channel	169
	8.1 The coding theorem	169
	8.2 The . - capacity	171
	8.3 The additivity problem	174
	8.3.1 The effect of entanglement in encoding and decoding	174
	8.3.2 A hierarchy of additivity properties	178
	8.3.3 Some entropy inequalities	180
	8.3.4 Additivity for complementary channels	183
	8.3.5 Nonadditivity of quantum entropy quantities	185
	8.4 Notes and references	192
	9 Entanglement-assisted classical communication	194
	9.1 The gain of entanglement assistance	194
	9.2 The classical capacities of quantum observables	198
	9.3 Proof of the Converse Coding Theorem	202
	9.4 Proof of the Direct Coding Theorem	204
	9.5 Notes and references	208
	10 Transmission of quantum information	209
	10.1 Quantum error-correcting codes	209
	10.1.1 Error correction by repetition	209
	10.1.2 General formulation	211
	10.1.3 Necessary and sufficient conditions for error correction	212
	10.1.4 Coherent information and perfect error correction	214
	10.2 Fidelities for quantum information	217
	10.2.1 Fidelities for pure states	217
	10.2.2 Relations between the fidelity measures	219