Jaeger G. Quantum Information: An Overview
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Preface
Quantum information science is a rapidly developing area of interdisciplinary
investigation at the nexus of quantum mechanics and information theory. It
now plays a significant role in a number of subdisciplines of physics, informa-
tion technology, and engineering. A number of books on quantum information
are available but are becoming outdated and/or differ significantly in approach
from this book, or cover only particular aspects of the subject. Historically,
lecture notes for the first general course on quantum information, given at
Hewlett-Packard and edited by Lo, Popescu, and Spiller, were published in
1998 [289]. Physicists have also long benefited from the generously provided
on-line lecture notes of Preskill [341]. At least one comprehensive monograph
on quantum information science was published at the turn of the century,
namely, the meticulous nearly 700-page book of Nielsen and Chuang [315].
More recent books by Paviˇci´c [325] and Stenholm and Suominen [404] are
noteworthy for their utility. A monograph detailing the mathematical founda-
tions of quantum information theory by Hayashi, which originally appeared
in Japanese in 2003, has just appeared in English [208]. The books of Benenti,
Casati, and Strini [33] and Gruska [201] are valuable textbooks for teaching
the subject. Quantum key distribution and quantum computing are currently
the most exciting applications of quantum information science that are suf-
ficiently well developed that a number of books are specifically dedicated to
one or the other of them: in the case of quantum cryptography and com-
munication, comprehensive collections have been edited by Alber et al. [7],
Beth and Leuchs [59], Bouwmeester, Ekert, and Zeilinger [72], Braunstein et
al. [79], and Sergienko [375]; in the case of quantum computation, the books
of Brylinski and Chen [89], Hirvensalo [216], Kitaev, Shen, and Vyalyi [252],
and Pittenger [335], and that edited by Lomonaco [287] focus on quantum
algorithms and/or associated mathematics. A number of popular books on
quantum computing have also been published, for example [83, 121].
In our information age, electronic access to primary sources is widely avail-
able, allowing one to locate the finest details of original investigations once one
is well oriented with tools and references in hand. Therefore, now what one of-
xii Preface
ten needs most when approaching the subject of quantum information science
is an overview that efficiently yet rigorously presents the fundamentals and
that provides a detailed weblinked bibliography to take one further [1]. This
book is intended to be such a handy reference for practitioners and students of
quantum physics and computer science that also treats foundational aspects
of quantum mechanics connected with quantum information science, includ-
ing those associated with quantum measurement which plays an essential role
in relating classical and quantum information. Most of the examples provided
here are quantum-optical ones as a pragmatic matter, arising from the fact
that interferometry is central to quantum information processing and the fact
that interferometry has primarily progressed through optical physics. How-
ever, exciting innovations have been made by experimental groups working
with a range of physical systems. Hopefully, workers in areas of experimental
physics and engineering other than optics will soon provide comprehensive
and detailed overviews of each of the experimental methods of manipulat-
ing quantum information. For the time being, discussions of various devices
for quantum information processing can be found [19, 74, 407]. Particularly
noteworthy are the books edited by Everitt [165] and Leggett et al. [274].
In the twentieth century, the formalism introduced in Dirac’s The Princi-
ples of Quantum Mechanics and von Neumann’s Mathematische Grundlagen
der Quantenmechanik was brought to bear on a broad range of physical prob-
lems. Elements of this formalism and related mathematics are outlined in the
appendices, together with standard quantum postulates. During the last two
decades of the twentieth century, investigations of the foundational problems
of quantum mechanics and the physics of computation were pivotal in giving
rise to quantum information science as a subject in its own right, providing
a conceptual basis for the development of quantum protocols and algorithms.
In turn, the investigation of foundational problems has benefited from the
work of those seeking solutions to central issues in quantum information sci-
ence, such as those of communication complexity. Aspects of this important
interplay have been addressed here.
It is my hope that, in addition to its serving as a practical tool for re-
searchers and students, this book will assist those seeking to understand the
subject to appreciate the many decades of work back to which the origins of
this exciting, relatively new field can be traced. Although the aim in includ-
ing this material is not to present a history of the exploration of foundations
of quantum mechanics or its philosophical underpinnings, a number of perti-
nent such results from earlier decades of the twentieth century are included
because they will likely prove important to future progress in both quantum
mechanics and information theory. The discussion of early work is here often
in the language of quantum information so as to facilitate access to earlier,
foundational work in quantum mechanics by those approaching fundamental
issues from a twenty-first century perspective.
Gregg Jaeger Cambridge, MA, August 2006
Acknowledgments
I gratefully acknowledge Paul Busch, Silvia Carrasco, John M. Myers, Sandu
Popescu, and Tommaso Toffoli for their very helpful suggestions during the
preparation of this book, my wife Savina Andonova Jaeger for her personal
support, my colleagues Millard Baublitz, Peter Busher, Gianni di Giuseppe,
Bahaa Saleh, Alexander Sergienko, and Malvin C. Teich for their professional
support, and my editors and the staff at Springer for their assistance. Last,
but not least, I acknowledge Abner Shimony as an inspiration to me, as to
many others seeking to better understand the subtleties of Nature.
Contents
Foreword ...................................................... vii
Preface ........................................................ xi
Acknowledgments .............................................xiii
1 Qubits ..................................................... 1
1.1 Quantumstate purity.................................... 5
1.2 The representation of qubits .............................. 8
1.3 Stokesparameters ....................................... 11
1.4 Single-qubitgates ....................................... 14
1.5 Thedouble-slit experiment ............................... 18
1.6 The Mach–Zehnder interferometer ......................... 23
1.7 Quantum coherence and information processing ............. 25
2 Measurements and quantum operations .................... 29
2.1 The von Neumann classification of processes ................ 32
2.2 ThePauli classification ofmeasurements ................... 34
2.3 Expectationvalues andthe vonNeumann projection......... 35
2.4 The L¨uders rule ......................................... 37
2.5 Reduced statisticaloperators.............................. 38
2.6 Generalquantum operations .............................. 39
2.7 Positive-operator-valued measures ......................... 41
3 Quantum nonlocality and interferometry .................. 45
3.1 Hiddenvariables andstate completeness.................... 46
3.2 VonNeumann’s “no-go”theorem.......................... 48
3.3 TheEinstein–Podolsky–Rosen argument.................... 49
3.4 Gleason’stheorem ....................................... 51
3.5 Bell inequalities ......................................... 52
3.6 Interferometric complementarity........................... 57
xvi Contents
3.7 TheFranson interferometer ............................... 61
3.8 Two-qubit quantumgates ................................ 63
4 Classical information and communication .................. 67
4.1 Communication channels ................................. 68
4.2 Shannonentropy ........................................ 70
4.3 R´enyientropy........................................... 74
4.4 Coding................................................. 74
4.5 Errorcorrection ......................................... 77
4.6 Data compression ....................................... 78
4.7 Communication complexity ............................... 79
5 Quantum information ...................................... 81
5.1 Quantumentropy ....................................... 82
5.2 Quantumrelative andconditional entropies................. 84
5.3 Quantummutual information ............................. 85
5.4 Fidelityand coherentinformation ......................... 86
5.5 Quantum R´enyi and Tsallis entropies ...................... 88
6 Quantum entanglement .................................... 91
6.1 Basic definitions......................................... 92
6.2 TheSchmidt decomposition............................... 94
6.3 Specialbases anddecompositions.......................... 95
6.4 Stokesparameters and entanglement....................... 98
6.5 Partialtranspose andreductioncriteria .................... 99
6.6 The“fundamental postulate” .............................101
6.7 Entanglement monotones.................................102
6.8 Distillation andboundentanglement.......................104
6.9 Entanglement andmajorization ...........................105
6.10 Concurrence ............................................106
6.11 Entanglement witnesses ..................................107
6.12 Entanglementas aresource ...............................108
6.13 Thethermodynamic analogy..............................109
6.14 Informationand thefoundationsof physics .................112
6.15 Thegeometry ofentanglement ............................114
6.16 Creatingentangled photons...............................115
7 Entangled multipartite systems ............................121
7.1 Stokesand correlationtensors.............................124
7.2 N-tangle................................................126
7.3 Generalized Schmidtdecomposition........................127
7.4 Lorentz-group isometries .................................127
7.5 Entanglement classes ....................................129
7.6 Algebraicinvariants of multipartitesystems.................131
7.7 Three-qubit statesand residualtangle......................133
Contents xvii
7.8 Three-qubit quantumlogicgates ..........................135
7.9 Statesof higherqubit number.............................136
8 Quantum state and process estimation .....................139
8.1 Quantumstate tomography...............................140
8.2 Quantum process tomography .............................143
8.3 Direct estimationmethods................................144
9 Quantum communication ..................................147
9.1 Quantumchannels.......................................148
9.2 Quantumchannel capacities ..............................149
9.3 Holevo’stheorem........................................151
9.4 Discrimination ofquantumstates..........................153
9.5 Theno-cloning theorem ..................................156
9.6 Basic quantumchannels..................................157
9.7 TheGHJW theorem.....................................159
9.8 Quantumdense coding...................................160
9.9 Quantumteleportation...................................162
9.10 Entanglement“swapping” ................................164
9.11 Entanglement“purification” ..............................165
9.12 Quantum data compression ...............................167
9.13 Quantumcommunicationcomplexity.......................169
10 Quantum decoherence and its mitigation ..................171
10.1 Quantumdecoherence....................................172
10.2 Decoherenceand mixtures................................173
10.3 Decoherence-free subspaces ...............................174
10.4 Quantumcoding,error detection, andcorrection ............175
10.5 Thenine-qubit Shorcode.................................179
10.6 Stabilizercodes .........................................181
10.7 Concatenationof quantumcodes ..........................183
11 Quantum broadcasting, copying, and deleting ..............185
11.1 Quantumbroadcasting...................................185
11.2 Quantumcopying .......................................186
11.3 Quantumdeleting .......................................189
11.4 Landauer’sprinciple .....................................190
12 Quantum key distribution .................................191
12.1 Cryptographyandcryptosystems..........................191
12.2 QKDsystems...........................................193
12.3 TheBB84 (four-state) protocol............................195
12.4 TheE91 (Ekert)protocol.................................197
12.5 TheB92 (two-state) protocol .............................198
12.6 Thesix-state protocol....................................199
xviii Contents
12.7 Eavesdropping ..........................................199
12.8 Securityproofs..........................................201
13 Classical and quantum computing .........................203
13.1 Classical computingandcomputational complexity ..........204
13.2 DeterministicTuring machines ............................206
13.3 Probabilistic Turing machines .............................207
13.4 Multi-tapeTuringmachines...............................208
13.5 QuantumTuringmachines................................209
13.6 Quantumcomputationalcomplexity .......................211
13.7 Fault-tolerantquantum computing.........................214
13.8 Linearopticalquantum computation.......................215
14 Quantum algorithms .......................................219
14.1 TheDeutsch–Jozsa algorithm .............................220
14.2 TheGrover search algorithm..............................221
14.3 TheShor factoringalgorithm .............................224
14.4 TheSimon algorithm ....................................229
A Mathematical elements ....................................231
A.1 Boolean algebraand Galoisfields .........................231
A.2 Random variables .......................................232
A.3 Vector Spaces andHilbertspace...........................233
A.4 The standard quantumformalism .........................237
A.5 The Dirac notation ......................................237
A.6 Groups oftransformations................................239
A.7 Probability, lattices, and posets ...........................240
A.8 Projectors, correlations,and theKochen–Specker theorem ....242
A.9 Traditional quantumlogic ................................243
B The quantum postulates ...................................245
B.1 Thestandard postulates..................................245
B.2 TheHeisenberg–Robertsonuncertainty relation .............247
B.3 Liouville space and open quantum systems ..................248
References .....................................................249
Index ..........................................................271
