Quantum Information Theory By Michael Aaron Nielsen DISSERTATION

260 Pages·2000·2.02 MB·English

Quantum Information Theory
By
Michael Aaron Nielsen
B.Sc., University of Queensland, 1993
B.Sc. (First Class Honours), Mathematics, University of Queensland, 1994
M.Sc., Physics, University of Queensland, 1998
DISSERTATION
Submitted in Partial Ful(cid:12)llment of the
Requirements for the Degree of
Doctor of Philosophy
Physics
The University of New Mexico
Albuquerque, New Mexico, USA
December 1998
(cid:13)c1998, Michael Aaron Nielsen
ii
Dedicated to the memory of Michael Gerard Kennedy
24 January 1951 { 3 June 1998
iii
Acknowledgments
It is a great pleasure to thank the many people who have contributed to this Dissertation. My
deepest thanks goes to my friends and family, especially my parents, Howard and Wendy, for
their support and encouragement. Warm thanks also to the many other people who have con-
tributed to this Dissertation, especially Carl Caves, who has been a terri(cid:12)c mentor, colleague,
and friend; to Gerard Milburn, who got me started in physics, in research, and in quantum in-
formation; and to Ben Schumacher, whose boundless enthusiasm and encouragement has provided
so much inspiration for my research. Throughout my graduate career I have had the pleasure
of many enjoyable and helpful discussions with Howard Barnum, Ike Chuang, Chris Fuchs, Ray-
mond Laflamme, Manny Knill, Mark Tracy, and Wojtek Zurek. In particular, Howard Barnum,
Carl Caves, Chris Fuchs, Manny Knill, and Ben Schumacher helped me learn much of what I know
about quantum operations, entropy, and distance measures for quantum information. The material
reviewedinchapters3through5Ilearntinnosmallmeasurefromthesepeople. Manyotherfriends
and colleagues have contributed to this Dissertation. A partial list includes Chris Adami, Dorit
Aharonov,JamesAnglin,DavidBeckman,PaulBenio(cid:11),CharlesBennett,SamBraunstein,Nicholas
Cerf, Ignacio Cirac, Richard Cleve, John Cortese, Vageli Coutsias, Wim van Dam, Ivan Deutsch,
DavidDiVincenzo,RobDuncan,MarkEttinger,BettyFry,JackGlassman,DanielGottesman,Bob
Gri(cid:14)ths,GaryHerling,TomHess,AlexanderHolevo,PeterH(cid:28)yer,RichardHughes,ChrisJarzynski,
BradJohnson,Je(cid:11)Kimble,AlexeiKitaev,Andrew Landahl,Debbie Leung,SethLloyd,Hoi-Kwong
Lo, Hideo Mabuchi, Eleanor Maes, Norine Meyer, Cesar Miquel, Juan Pablo Paz, Mitch Porter,
John Preskill,Peter Shor, JohnSmolin, Alain Tapp, BarbaraTerhal, Amnon Ta-Shma, Mike West-
moreland,BarbyWoods,BillWootters,ChristofZalka,andPeterZoller. Myespecialthanksalsoto
thosecreativepeoplewithoutwhomthephysicsofinformationwouldnotexistasa(cid:12)eldofresearch;
especial influences on my thinking were the writings of Paul Benio(cid:11), Charles Bennett, Gilles Bras-
sard, David Deutsch, Artur Ekert, Richard Feynman, Alexander Holevo, Richard Jozsa, Rolf Lan-
dauer, Go¨ran Lindblad, Seth Lloyd, Asher Peres, Ben Schumacher, Peter Shor, John Wheeler,
Stephen Wiesner, Bill Wootters, and Wojtek Zurek.
Michael Aaron Nielsen
The University of New Mexico
December 1998
iv
Quantum Information Theory
By
Michael Aaron Nielsen
Abstract of Dissertation
Submitted in Partial Ful(cid:12)llment of the
Requirements for the Degree of
Doctor of Philosophy
Physics
The University of New Mexico
Albuquerque, New Mexico, USA
December 1998
v
Quantum Information Theory
By
Michael Aaron Nielsen
B.Sc., University of Queensland, 1993
B.Sc. (First Class Honours), Mathematics, University of Queensland, 1994
M.Sc., Physics, University of Queensland, 1998
Abstract
Quantum information theory is the study of the achievable limits of information processing within
quantum mechanics. Many di(cid:11)erent types of information can be accommodated within quantum
mechanics, including classical information, coherent quantum information, and entanglement. Ex-
ploring the rich variety of capabilities allowed by these types of information is the subject of quan-
tum information theory, and of this Dissertation. In particular, I demonstrate several novel limits
to the information processing ability of quantum mechanics. Results of especial interest include:
the demonstrationof limitations to the class of measurements which may be performed in quantum
mechanics; a capacity theorem giving achievable limits to the transmission of classical information
through a two-way noiseless quantum channel; resource bounds on distributed quantum compu-
tation; a new proof of the quantum noiseless channel coding theorem; an information-theoretic
characterizationof the conditions under which quantum error-correctionmay be achieved; an anal-
ysis ofthe thermodynamic limits to quantumerror-correction,andnew bounds onchannelcapacity
for noisy quantum channels.
vi
Contents
List of figures viii
Nomenclature and notation ix
I Fundamentals of quantum information 1
1 The physics of information 3
1.1 A collision of ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 What observables are realizable as quantum measurements? . . . . . . . . . . . . . 5
1.3 Overview of the (cid:12)eld of quantum information . . . . . . . . . . . . . . . . . . . . . 8
1.4 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Quantum information, science, and technology . . . . . . . . . . . . . . . . . . . . . 12
2 Quantum information: fundamentals 19
2.1 Quantum bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Superdense coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Quantum teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 What quantum measurements may be realized? . . . . . . . . . . . . . . . . . . . . 28
2.6 Experimental quantum information processing . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Proposals for quantum information processing . . . . . . . . . . . . . . . . 31
2.6.2 Experimental demonstration of quantum teleportation using NMR . . . . . 36
3 Quantum operations 41
3.1 Quantum operations: fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Quantum operations on a single qubit . . . . . . . . . . . . . . . . . . . . . 50
3.2 Freedom in the operator-sum representation . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Teleportation as a quantum operation. . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Quantum process tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 One qubit example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 The POVM formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Beyond quantum operations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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4 Entropy and information 66
4.1 Shannon entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Basic properties of entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 The binary entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 The relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Mutual information and conditional entropy . . . . . . . . . . . . . . . . . 69
4.2.4 The data processing inequality . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Von Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Quantum relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Basic properties of entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.3 Measurements and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.4 The entropy of ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.5 Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.6 Concavity of the entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Strong subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Distance measures for quantum information 89
5.1 Distance measures for classical information . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 How close are two quantum states? . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Absolute distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.2 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.3 Distance measures derived from (cid:12)delity . . . . . . . . . . . . . . . . . . . . 98
5.2.4 Relationships between distance measures . . . . . . . . . . . . . . . . . . . 99
5.3 Dynamic measures of information preservation . . . . . . . . . . . . . . . . . . . . . 100
5.3.1 Continuity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.2 Chaining quantum errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Alternative view of the dynamic measures . . . . . . . . . . . . . . . . . . . . . . . 105
II Bounds on quantum information transmission 108
6 Quantum communication complexity 110
6.1 The Holevo bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Capacity theorem for qubit communication . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Communication complexity of the inner product . . . . . . . . . . . . . . . . . . . . 119
6.3.1 Converting exact protocols into clean form . . . . . . . . . . . . . . . . . . 120
6.3.2 Reduction from the communication problem . . . . . . . . . . . . . . . . . 121
6.3.3 Lower bounds for bit protocols . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4 Coherent quantum communication complexity . . . . . . . . . . . . . . . . . . . . . 122
6.4.1 Coherent communication complexity of the quantum Fourier transform . . 123
6.4.2 A general lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 A uni(cid:12)ed model for communication complexity. . . . . . . . . . . . . . . . . . . . . 127
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7 Quantum data compression 130
7.1 Schmidt numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 Typical subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3 Quantum data compression theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 136
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7.4 Quantum data compression with a classical side channel . . . . . . . . . . . . . . . 139
7.5 Universal data compression with a classical side channel . . . . . . . . . . . . . . . 141
7.5.1 A dense subset of density operators with distinct entropies . . . . . . . . . 143
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8 Entanglement 147
8.1 Pure state entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2 Mixed state entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3 Entanglement: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9 Error correction and Maxwell’s demon 163
9.1 Entropy exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.2 Quantum Fano inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.3 The quantum data processing inequality . . . . . . . . . . . . . . . . . . . . . . . . 167
9.4 Quantum error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.4.1 Shor’s code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9.5 Information-theoretic conditions for error correction . . . . . . . . . . . . . . . . . . 175
9.6 Information-theoretic inequalities for quantum processes . . . . . . . . . . . . . . . 179
9.7 Quantum error correction and Maxwell’s demon . . . . . . . . . . . . . . . . . . . . 182
9.7.1 Error-correctionby a \Maxwell demon" . . . . . . . . . . . . . . . . . . . . 182
9.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10 The quantum channel capacity 189
10.1 Noisy channel coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.2 Classical noisy channels in a quantum setting . . . . . . . . . . . . . . . . . . . . . 191
10.3 Coherent information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.3.1 Properties of coherent information . . . . . . . . . . . . . . . . . . . . . . . 193
10.3.2 The entropy-(cid:12)delity lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.3.3 Quantum characteristics of the coherent information I . . . . . . . . . . . . 195
10.3.4 Quantum characteristics of the coherent information II . . . . . . . . . . . 197
10.4 Noisy channel coding revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.4.1 Mathematical formulation of noisy channel coding . . . . . . . . . . . . . . 199
10.5 Upper bounds on the channel capacity . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.5.1 Unitary encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.5.2 General encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.5.3 Other encoding protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.7 Channels with a classical observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.7.1 Upper bounds on channel capacity . . . . . . . . . . . . . . . . . . . . . . . 210
10.7.2 Relationship to unobserved channel . . . . . . . . . . . . . . . . . . . . . . 213
10.7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
10.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
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III Conclusion 218
11 Conclusion 220
11.1 Summary of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.2 Open problems in quantum information. . . . . . . . . . . . . . . . . . . . . . . . . 222
11.2.1 A unifying picture for quantum information. . . . . . . . . . . . . . . . . . 223
11.2.2 Classical physics and the decoherence program . . . . . . . . . . . . . . . . 225
11.2.3 Quantum information and statistical physics . . . . . . . . . . . . . . . . . 226
11.3 Concluding thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A Purifications and the Schmidt decomposition 228
Bibliography 232
Index 245
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