Representations of quantum operations, with applications to quantum cryptography

157 Pages·2004·0.96 MB·English

by  Pablo J. Arrighi

Technical Report
UCAM-CL-TR-595
ISSN 1476-2986
Number 595
Computer Laboratory
Representations of quantum
operations, with applications to
quantum cryptography
Pablo J. Arrighi
July 2004
15 JJ Thomson Avenue
Cambridge CB3 0FD
United Kingdom
phone [phone]
http://www.cl.cam.ac.uk/
c 2004 Pablo J. Arrighi
(cid:13)
This technical report is based on a dissertation submitted 23
September 2003 by the author for the degree of Doctor of
Philosophy to the University of Cambridge, Emmanuel College.
Technical reports published by the University of Cambridge
Computer Laboratory are freely available via the Internet:
http://www.cl.cam.ac.uk/TechReports/
ISSN 1476-2986
Abstract
Representations of quantum operations
We start by introducing a geometrical representation (real vector space) of quantum states
andquantumoperations. Todosoweexploitanisomorphismfrompositivematricestoasub-
coneoftheMinkowskifuturelight-cone. Purestatesmapontocertainlight-likevectors,whilst
the axis of revolution encodes the overall probability of occurrence for the state. This exten-
sion of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum
operations, and in particular to view the per-outcome effects of generalized measurements.
We show that these consist of the product of an orthogonal transform about the axis of the
cone of revolution and a positive real symmetric linear transform. In the case of a qubit the
representation becomes all the more interesting since it elegantly associates, to each measure-
ment element of a generalized measurement, a Lorentz transformation in Minkowski space.
We formalize explicitly this correspondence between ‘observation of a quantum system’ and
‘special relativistic change of inertial frame’. To end this part we review the state-operator
correspondence, which was successfully exploited by Choi to derive the operator-sum repre-
sentation of quantum operations. We go further and show that all of the important theorems
concerningquantumoperationscaninfactbederivedassimplecorollariesofthoseconcerning
quantum states. Using this methodology we derive novel composition laws upon quantum
states and quantum operations, Schmidt-type decompositions for bipartite pure states and
some powerful formulae relating to the correspondence.
Quantum cryptography
The key principle of quantum cryptography could be summarized as follows. Honest par-
ties communicate using quantum states. To the eavesdropper these states are random and
non-orthogonal. In order to gather information she must measure them, but this may cause
irreversible damage. Honest parties seek to detect her mischief by checking whether certain
quantum states are left intact. Thus tradeoff between the eavesdropper’s information gain,
and the disturbance she necessarily induces, can be viewed as the power engine behind quan-
tum cryptographic protocols. We begin by quantifying this tradeoff in the case of a measure
distinguishing two non-orthogonal equiprobable pure states. A formula for this tradeoff was
first obtained by Fuchs and Peres, but we provide a shorter, geometrical derivation (within
the framework of the above mentioned conal representation). Next we proceed to analyze the
Information gain versus disturbance tradeoff in a scenario where Alice and Bob interleave, at
random, pairwise superpositions of two message words within their otherwise classical com-
munications. Thisworkconstitutesoneofthefewresultscurrentlyavailableregardingd-level
systems quantum cryptography, and seems to provide a good general primitive for building
such protocols. The proof crucially relies on the state-operator correspondence formulae de-
rived in the first part, together with some methods by Banaszek. Finally we make use of this
analysis to prove the security of a ‘blind quantum computation’ protocol, whereby Alice gets
Bob to perform some quantum algorithm for her, but prevents him from learning her input
to this quantum algorithm.
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Acknowledgements
I am very grateful to my superviser, Dr. Anuj Dawar, who has never failed his role. For some
good advice and within the Computer Laboratory still, I would like to thank Prof. Glynn
Winskel and Dr. Alan Mycroft. I am also very appreciative of my collaborators: Dr. Louis
Salvail, Prof. Frank Kelly on one occasion, and above all Christophe Patricot with whom
−
thoughts met friendship in an exceptional manner.
A small number of great friends made my stay in Cambridge into extremely happy days:
special thanks to S´ebastien Loisel, Christophe Patricot, Sandra Lucas, Emilie Stephenson
and Tom Kelly. I am also indebted to my uncles: Dr. Arturo Lezama made me discover
quantum theory when I was little, Dr. Jean-Michel Arrighi provided encouragements and
financial support throughout my studies, together with Dr. Paul Arrighi and my two grand-
fathers. Parentsareneverthankedenough. Ithankmyfather,PierreArrighi,forthediagrams
in this thesis and so many other things. I thank my mother, Claudia Lezama, for the title of
chapter 2 and so many other things. I am also grateful to Jacques Coral and Ilse Arrighi.
I thank my sisters, Ga¨ıa Lezama, Barbara Arrighi, Malena Arrighi who still love me in spite
of the well-known series of events who took me to England, and away from their childhoods.
Thank you Elvire, for this last year and those ahead of us.
4
Collaborations and Publications
Chapter 2 has been published under the title
•
Quantum Computation explained to my Mother
in the Bulletin of the EATCS, 80, 134-142, (2003), and in Current Trends in Theoreti-
cal Computer Science, Ed. G. Paun, G. Rozenberg, World Scientific, (2004). It is also
available at arXiv:quant-ph/0212062.
TheresultsinChapter3andChapter6havebeenacceptedforpublicationasonesingle
•
article under the title
The Conal representation of Quantum States and Non
Trace-Preserving Quantum Operations
in Phys. Rev. A, 68, 042310, (2003), and in Virtual Journal on Quantum Information,
3, 10. They are also available at arXiv:quant-ph/0212062.
This work is the outcome of a collaboration with Christophe Patricot
from the DAMTP, Faculty of Mathematics, University of Cambridge, UK.
The results in Chapter 4 have been published under the title
•
A Note on the Correspondence between Qubit Quantum
Operations and Special Relativity
in J. Phys. A, 36, L287-L296, (2003), and are available at arXiv:quant-ph/0212135.
This work is the outcome of a collaboration with Christophe Patricot.
The results in Chapter 5 have been published under the title
•
On Quantum Operations as Quantum States
in Annals of Phys. 311, 26-52, (2004), and are available at arXiv:quant-ph/0307024.
This work is the outcome of a collaboration with Christophe Patricot.
The results in Chapter 7 have been published under the title
•
Quantum Decoys
in Int. J. Quantum Information, and are available at arXiv:quant-ph/0308050.
The results in Chapter 8 are available under the title
•
Blind Quantum Computation
at arXiv:quant-ph/0309152.
This work is the outcome of a collaboration with Dr. Louis Salvail
from BRICS, Department of Computer Science, University of Aarhus, Denmark. More-
over
Lemma 8.2 is due to Prof. Frank Kelly from the DPMMS, Faculty of Math-
ematics, University of Cambridge, UK.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Collaborations and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Introduction 11
1 Overture 15
1.1 Scope and aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 How to read the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Quantum computation explained to my mother 27
2.1 Some mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.3 Matrices of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 Matrices of complex numbers . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.5 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 The quantum setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.4 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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I Representations of quantum operations 41
3 Conal representation 43
3.1 Conal representation of d-dimensional quantum systems . . . . . . . . . . . . 44
3.1.1 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Generalized density matrices . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.3 Generalized measurements . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.4 Quantum operations represented in the cone . . . . . . . . . . . . . . 50
3.2 The Qubit case pushed further . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 The cone and the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 Post-measurement states. . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.3 Square and square root . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Inner products through quantum operations . . . . . . . . . . . . . . . 56
3.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Qubit quantum operations and special relativity 59
4.1 Quantum operations as Lorentz transforms and vice-versa . . . . . . . . . . . 64
4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 On quantum operations as quantum states 73
5.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.2 Useful formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1.3 The correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Properties of quantum states and quantum operations . . . . . . . . . . . . . 85
5.2.1 Properties rediscovered via the correspondence . . . . . . . . . . . . . 85
5.2.2 Properties discovered via the correspondence . . . . . . . . . . . . . . 90
5.2.3 Trace-preserving quantum operations. . . . . . . . . . . . . . . . . . . 93
5.3 Induced geometrical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 Composition laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.2 Duality: states and functionals . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 103
II Quantum cryptography 107
6 The qubit information gain versus disturbance 109
6.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.1 Information contribution. . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.2 Disturbance contribution . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Optimization and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Quantum decoys 117
7.1 Mathematical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Preliminary calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2.1 Information gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2.2 Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Optimization and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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8 Blind quantum computation 127
8.1 Principles of a solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Information gain versus disturbance tradeoff . . . . . . . . . . . . . . . . . . . 131
8.3 Protocol and Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Conclusion 139
9 Achievements and Further research 141
9.1 On quantum theory with real vector spaces . . . . . . . . . . . . . . . . . . . 141
9.2 On quantum operations as quantum states . . . . . . . . . . . . . . . . . . . . 142
9.3 On information gain versus disturbance tradeoffs . . . . . . . . . . . . . . . . 143
9.4 On blind quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A Notation 147
A.1 Common formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.2 Formalism specific to Chapters 3, 4 and 6 . . . . . . . . . . . . . . . . . . . . 150
A.3 Formalism specific to Chapters 5 and 7. . . . . . . . . . . . . . . . . . . . . . 151
List of Figures
The conal representation of a qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Optimal measurement family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Decoys’ information gain versus disturbance curves . . . . . . . . . . . . . . . . . . 125
List of Table(s)
Summary of the state-operator correspondence . . . . . . . . . . . . . . . . . . . . 104
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