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 Fulfillment of the Requirements for the Degree of Doctor of Philosophy Physics The University of New Mexico arXiv:quant-ph/0011036v1 9 Nov 2000 Albuquerque, New Mexico, USA December 1998 c 1998, 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 terrific 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 reviewed in chapters 3 through 5 I learnt in no small measure from these people. Many other friends and colleagues have contributed to this Dissertation. A partial list includes Chris Adami, Dorit Aharonov, James Anglin, David Beckman, Paul Benioff, Charles Bennett, Sam Braunstein, Nicholas Cerf, Ignacio Cirac, Richard Cleve, John Cortese, Vageli Coutsias, Wim van Dam, Ivan Deutsch, David DiVincenzo, Rob Duncan, Mark Ettinger, Betty Fry, Jack Glassman, Daniel Gottesman, Bob Griffiths, Gary Herling, Tom Hess, Alexander Holevo, Peter Høyer, Richard Hughes, Chris Jarzynski, Brad Johnson, Jeff Kimble, Alexei Kitaev, Andrew Landahl, Debbie Leung, Seth Lloyd, Hoi-Kwong Lo, Hideo Mabuchi, Eleanor Maes, Norine Meyer, Cesar Miquel, Juan Pablo Paz, Mitch Porter, John Preskill, Peter Shor, John Smolin, Alain Tapp, Barbara Terhal, Amnon Ta-Shma, Mike West- moreland, Barby Woods, Bill Wootters, Christof Zalka, and Peter Zoller. My especial thanks also to those creative people without whom the physics of information would not exist as a field of research; especial influences on my thinking were the writings of Paul Benioff, Charles Bennett, Gilles Bras- sard, David Deutsch, Artur Ekert, Richard Feynman, Alexander Holevo, Richard Jozsa, Rolf Lan- dauer, G¨oran 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 Fulfillment 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 different 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 demonstration of 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 characterization of the conditions under which quantum error-correction may be achieved; an anal- ysis of the thermodynamic limits to quantum error-correction, and new bounds on channel capacity for noisy quantum channels. vi Contents List of figures xi Nomenclature and notation xiii I Fundamentals of quantum information 1 1 The physics of information 3 1.1 Acollisionofideas ............................... ..... 3 1.2 What observables are realizable as quantum measurements? ............. 5 1.3 Overview of the field of quantum information . .......... 8 1.4 OverviewoftheDissertation . ........ 9 1.5 Quantum information, science, and technology . ............. 12 2 Quantum information: fundamentals 19 2.1 Quantumbits ..................................... .. 19 2.2 Superdensecoding................................ ..... 21 2.3 Quantumteleportation .. .. .. .. .. .. .. .. .. .. .. .. ...... 23 2.4 Quantumcomputation .............................. .... 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 teleportationusingNMR. 36 3 Quantum operations 41 3.1 Quantumoperations: fundamentals . ......... 43 3.1.1 Quantumoperationsonasinglequbit . ...... 50 3.2 Freedom in the operator-sumrepresentation . ............ 51 3.3 Teleportationasaquantumoperation. .......... 54 3.4 Quantumprocesstomography . ...... 57 3.4.1 Onequbitexample ............................... 59 3.5 ThePOVMformalism ................................ .. 62 3.6 Beyondquantumoperations? . ...... 63 vii 4 Entropy and information 66 4.1 Shannonentropy .................................. ... 66 4.2 Basicpropertiesofentropy. ........ 67 4.2.1 Thebinaryentropy .............................. 67 4.2.2 Therelativeentropy. .. .. .. .. .. .. .. .. .. .. .. ... 68 4.2.3 Mutual information and conditional entropy . ......... 69 4.2.4 Thedataprocessinginequality . ...... 72 4.3 VonNeumannentropy ............................... ... 73 4.3.1 Quantumrelativeentropy. .... 74 4.3.2 Basicpropertiesofentropy . ..... 75 4.3.3 Measurementsandentropy . ... 76 4.3.4 Theentropyofensembles . ... 77 4.3.5 Subadditivity................................. .. 78 4.3.6 Concavityoftheentropy . ... 79 4.4 Strongsubadditivity ............................. ...... 80 5 Distance measures for quantum information 89 5.1 Distance measures for classical information . .............. 90 5.2 Howclosearetwoquantumstates? . ....... 92 5.2.1 Absolutedistance .............................. .. 92 5.2.2 Fidelity...................................... 96 5.2.3 Distance measures derived from fidelity . ........ 98 5.2.4 Relationships between distance measures . ......... 99 5.3 Dynamic measures of information preservation . ............. 100 5.3.1 Continuityrelations . .... 104 5.3.2 Chainingquantumerrors . .. 104 5.4 Alternative view of the dynamic measures . .......... 105 II Bounds on quantum information transmission 108 6 Quantum communication complexity 110 6.1 TheHolevobound.................................. .. 111 6.2 Capacity theorem for qubit communication . ........... 116 6.3 Communication complexity of the inner product . ............ 118 6.3.1 Converting exact protocols into clean form . ......... 119 6.3.2 Reduction from the communication problem . ....... 120 6.3.3 Lowerboundsforbitprotocols . ..... 121 6.4 Coherent quantum communication complexity . ........... 122 6.4.1 Coherent communication complexity of the quantum Fourier transform . 122 6.4.2 Agenerallowerbound ............................ 124 6.5 A unified model for communication complexity . ........... 126 6.6 Conclusion...................................... .. 128 7 Quantum data compression 130 7.1 Schmidtnumbers .................................. .. 131 7.2 Typicalsubspaces................................ ..... 133 7.3 Quantumdatacompressiontheorem . ....... 136 viii 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 Purestateentanglement . .. .. .. .. .. .. .. .. .. .. .. ...... 148 8.2 Mixedstateentanglement . ...... 149 8.3 Entanglement:Examples . ...... 152 8.4 Conclusion...................................... .. 161 9 Error correction and Maxwell’s demon 163 9.1 Entropyexchange ................................. .... 164 9.2 QuantumFanoinequality. ...... 166 9.3 The quantum data processing inequality . .......... 167 9.4 Quantumerrorcorrection . ...... 169 9.4.1 Shor’scode.................................... 173 9.5 Information-theoretic conditions for error correction.................. 175 9.6 Information-theoretic inequalities for quantum processes ............... 178 9.7 Quantum error correction and Maxwell’s demon . ........... 181 9.7.1 Error-correctionbya“Maxwelldemon” . ....... 181 9.7.2 Discussion ...................................