Table Of ContentTopics in Applied Physics
Volume 102
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Topics in Applied Physics
Topics in Applied Physics is a well-established series of review books, each of which presents a
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Hiroshi Imai Masahito Hayashi (Eds.)
Quantum Computation
and Information
From Theory to Experiment
With49Figures
123
HiroshiImai
GraduateSchoolofInformation,ScienceandTechnology
TheUniversityofTokyo
7-3-1Hongo,Bunkyo-ku
Tokyo,113-8656Japan
and
ERATOQuantumComputationandInformationProject
JapanScienceandTechnologyAgency
201DainiHongoWhiteBldg
5-28-3,Hongo,Bunkyo-ku
Tokyo113-0033,Japan
[email protected]
MasahitoHayashi
ERATOQuantumComputationandInformationProject
JapanScienceandTechnologyAgency
201DainiHongoWhiteBldg
5-28-3,Hongo,Bunkyo-ku
Tokyo113-0033,Japan
[email protected]
LibraryofCongressControlNumber:2006923435
PhysicsandAstronomyClassificationScheme(PACS):
03.67.Lx,03.67.-a,03.67.Dd,03.67.Mn,42.50.-p
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ISSNelectronicedition:1437-0859
ISBN-10 3-540-33132-8SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-33132-2SpringerBerlinHeidelbergNewYork
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Preface
Once the quantum effect had been regarded as an obstacle to suitable in-
formation processing in existing information systems. Recently, it has been
discovered that quantum effect is, to the contrary, very useful as a resource
used in information processesing. This research field is called quantum infor-
mation and is rapidly growing as a new paradigm for information systems.
For example, we can factorize a large number quickly by Shor’s algorithm
on a quantum computer once a quantum computer is available, and we can
communicate securely without any assumption for computation complexity
by using quantum key distribution. These quantum information protocols
cannot be realized without quantum effects.
In the research of existing information processes, it is possible to study
hardware and software separately because their roles are clearly divided.
However, such separation between them becomes an obstacle for the whole
researchonquantuminformation.Towardthedevelopmentofquantumalgo-
rithmsandprotocols,itisnecessarytounderstandthemathematicaldescrip-
tionofquantumphenomena.Therealizationofquantuminformationsystems
requires development of quantum devices, for which we need to understand
the theoretical scheme of quantum information science. Therefore, we need
collaboration over the existing framework. To promote such collaboration,
we bring this book as a collection of overviews of selected topics in quantum
information.
Thisbookorganizedasfollows.Weexplainthepowerofquantumcompu-
tation in Part I. Currently, only Shor’s factorization algorithm and Grover’s
search algorithm are known to be faster on a quantum computer than on a
classical computer. The ability of a quantum computer cannot be cleared up
onlybydiscoveryofthesealgorithms.Now,manyresearchersareattempting
to developing better quantum devices to build a quantum computer. How-
ever, the research for the power of quantum computers is as important as
the research for quantum devices. Part I reviews the so-called identification
problem of an unknown function f using a quantum computer, where the
function f is often called an oracle. In fact, many problems in computer sci-
enceareformulatedinthisform.Forinstance,Grover’ssearchproblemisalso
in this form. Part I discusses the superiority of a quantum computer over a
classical computer for this type of problems. In particular, the Chapter by
Ambainis et al. treats the case of no error in the computation process, and
VI Preface
theChapterbyIwamaetal.coversthecasewheresomeerrorshappeninthe
specific points. By reviewing these topics, Part I signifies the importance of
building a quantum computer.
Part II focuses on the bounds of the power of several quantum informa-
tionprocessesandquantumentanglement,whichisanimportantresourcefor
quantum information protocols. TheChapter byHayashi dealswith theoret-
ical issues on the identification of the density matrix of a quantum system.
Since the perfect cloning of a quantum state is impossible and any mea-
surement demolishes quantum states, precise identification requires a better
measurementextractingmuchinformationfromthequantumsystem.Hence,
theselectionofmeasurementisanimportantissueofthistopic.Ontheother
hand, an approximate cloning is possible. The Chapter by Fan discusses the
boundoftheperformanceofthequantumapproximatecloning.Throughthe
Chapters by Hiroshima et al. and Matsumoto, we give an overview of the
researchonquantumentanglement.TheChapterbyHiroshimaetal.reviews
approaches toward quantum entanglement from various viewpoints. In par-
ticular, entanglement is closely related to the problem of sending a quantum
state via a noisy quantum channel. Such a relation is also discussed. The
Chapter by Matsumoto focuses on the additivity problem, the hottest topic
in quantum entanglement. This problem is essentially linked to the problem
on sending classical information via a noisy quantum channel. We highlight
thisconnection.Notethattheproblemonsendingquantumstateisdifferent
from the problem on sending classical information.
PartIIItreatssecurequantuminformationprocesses.Shor’sfactorization
algorithmmakestheRSApublic-keycryptosysteminsecureonceonebuildsa
quantumcomputer.Hence,wehavetopreparealternativecryptosystemsasa
countermeasureforrealizationofquantumcomputer.Oneideaisthedevelop-
ment of public-key cryptosystem that is secure even for quantum computers.
Another is an information-theoretically secure cryptographic system whose
security does not depend on the assumption for computational complexity.
The Chapter by Kawachi et al. highlights the former type of cryptosystems
by discussing the concept of one-way functions, which is a basic concept for
public key cryptosystems. Based on this concept, the quantum public-key
cryptosystem is explained. This cryptosystem well works on the assump-
tion that all component parts (eavesdropper, channel, sender, and receiver)
are quantum. The Chapter by Wang treats an information-theoretically se-
cure protocol that distributes a secret key via a quantum channel, which is
called quantum key distribution. Perfect single photons and noiseless quan-
tumchannelsarenecessaryfortherealizationoftheinitialprotocolproposed
byBennettandBrassard.Hence,weneedtoconsidertheprotocolofsending
imperfectsingle-photonsvianoisyandlossyquantumchannels.Thesecurity
oftheaboverealisticprotocolisthemaintopicofthischapter.Secureproto-
cols are not limited in cryptography. Steganography is known as a protocol
that keeps the secret of the existence of the communication. The Chapter by
Natori shows that quantum steganography exceeds classical steganography.
Preface VII
Finally, Part IV reports the research activities of realization of quantum
informationsystems.Thispartcontainsexperimentsconcerningquantumkey
distribution,apartofShor’sfactorizationalgorithm,andgenerationofentan-
gled states. First, we review 150km transmission quantum key distribution
and quantum key distribution with a real optical fiber of commercial use for
14 days. Next, we see how to realize quantum computation with 1024 qubits
of a part of Shor’s factorization algorithm. High-quality generation of entan-
gled states is also discussed in this part.
This book is organized so that each chapter can be read independently.
We recommend that the reader begins with the chapter of interest and then
expandtherangeofthisinterest.Wehopethatthereaderofthisbookwould
get interested in a wide research area of quantum information science.
In fact, the contents of this book mainly consist of research results
obtained by the ERATO Quantum Computation and Information (QCI)
Project. This project started in October 2000 by gathering interdiscipnary
researchers from various research fields as one of Exploratory Research for
Advanced Technology (ERATO) programs of Japan Science and Technol-
ogy Agency (JST). This project finished in September 2005, and continued
another program, Solution-Oriented Research for Science and Technology
(SORST)ofJST.Eachchapterofthisbookiswrittenbytheresearchersand
a visiting researcher of this project.
WewouldliketoexpressourgratitudetoMr.HideoOhgata,Mr.Jun-ichi
Hoshi,Mr.SatoshiAsada,andMr.TakanoriKamei,DepartmentofResearch
Project in JST, for their kind management. We are also thankful to all the
contributors for their interesting research manuscripts. We are also grate-
fultoalltheresearchersofERATOQuantumComputationandInformation
Projectandtheircollaborators.Moreover,weareparticularlyindebtedtoour
administrativeandsupportingstaff,Mr.MichiyukiAmaike,Ms.EmiBandai,
Ms. Miho Inagaki, Ms. Chie Matsumoto, Ms. Minako Ooyama, Ms. Takako
Sakuragi,Ms.HirokoTakeshima,andMr.NobuyoshiUmezawafortheirkind
support. Finally, we wish to thank Dr. Claus E. Ascheron of Springer-Verlag
forhisexcellentmanagementforthepublicationofthisbookandhisencour-
angement.
Hongo, Tokyo, Hiroshi Imai
January 2006 Masahito Hayashi
Contents
Part I Quantum Computation
Quantum Identification of Boolean Oracles
Andris Ambainis, Kazuo Iwama, Akinori Kawachi, Rudy Raymond,
Shigeru Yamashita .............................................. 3
1 Introduction ................................................ 3
2 Formalization............................................... 5
3 General Upper Bounds....................................... 8
4 Relation With Learning Theory ............................... 11
5 Tight Upper Bounds for Small M ............................. 12
6 Classical Lower and Upper Bounds ............................ 14
7 Concluding Remarks......................................... 15
References ..................................................... 16
Index.......................................................... 18
Query Complexity of Quantum Biased Oracles
Kazuo Iwama, Rudy Raymond, Shigeru Yamashita ................. 19
1 Introduction ................................................ 19
2 Goldreich–Levin Problem and Biased Oracles ................... 21
2.1 The Model of Quantum Biased Oracles .................... 26
3 Upper Bounds of the Query Complexity of Biased Oracles With
Special Conditions........................................... 27
3.1 Basic Tools for Quantum Computation .................... 27
3.2 Quantum Biased Oracles With the Same Bias Factor........ 28
3.3 Quantum Biased Oracles With Resettable Condition ........ 33
4 Lower Bounds of the Query Complexity of Biased Oracles ........ 34
5 Concluding Remarks......................................... 39
References ..................................................... 40
Index.......................................................... 42
Part II Quantum Information
Quantum Statistical Inference
Masahito Hayashi .............................................. 45
X Contents
1 Introduction ................................................ 45
2 Quantum State Estimation ................................... 47
2.1 State Estimation in Pure State Family .................... 47
2.2 State Estimation for Covariant Pure States Family.......... 48
2.3 State Estimation in Gaussian States Family ................ 49
2.4 State Estimation in Nonregular Family .................... 51
2.5 Estimation of Eigenvalue of Density Matrix in Qubit System . 52
3 Estimation of SU(2) Action With Entanglement................. 52
3.1 One-Parameter Case .................................... 53
3.2 Three-Parameter Case................................... 53
4 Hypothesis Testing and Discrimination......................... 54
4.1 Hypothesis Testing of Entangled State..................... 54
4.2 Distinguishability and Indistinguishability by LOCC ........ 55
4.3 Application of Quantum Hypothesis Testing................ 55
5 Experimental Application of Quantum Statistical Inference ....... 56
5.1 State Estimation in the Two-Qubit System ................ 57
5.2 Testing of Entangled State in the SPDC System ............ 57
6 Analysis on Quantum Measurement ........................... 58
6.1 Quantum Measurement With Negligible State Demolition.... 58
6.2 Quantum Universal Compression ......................... 58
References ..................................................... 59
Index.......................................................... 61
Quantum Cloning Machines
Heng Fan ...................................................... 63
1 Introduction ................................................ 63
2 Buˇzek and Hillery Universal Quantum Cloning Machine.......... 63
3 N to M UQCM (Gisin and Massar) ............................ 65
4 Universal Quantum Cloning Machine for General d-Dimensional
System, Werner Cloning Machine.............................. 65
5 A UQCM for d-Dimensional Quantum State Proposed by Fan et al. 66
6 Further Results About the UQCM............................. 67
6.1 UQCM for 2-Level System ............................... 68
6.2 UQCM for d-Level System ............................... 71
7 UQCM Realized in Real Physical Systems ...................... 74
8 UQCM for Identical Mixed States ............................. 79
8.1 A 2 to 3 Universal Quantum Cloning for Mixed States....... 79
8.2 General 2 to M(M >2) UQCM .......................... 81
9 Phase-Covariant Quantum Cloning Machine .................... 81
10 Transformation ............................................. 82
11 Hilbert–Schmidt Norm ....................................... 84
12 Bures Fidelity .............................................. 88
13 Quantum Cloning for x−y Equatorial Qubits .................. 90
14 Quantum Cloning Networks for Equatorial Qubits ............... 91
15 Separability of Copied Qubits and Quantum Triplicators ......... 93
