Asymptotic theory of quantum statistical inference

553 Pages·2005·3.149 MB·English

by  Masahito Hayashi

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ASYMPTOTIC THEORY OF QUANTUM STATISTICAL INFERENCE
Selected Papers
Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
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PREFACE
In orderto obtaininformationfroma quantumsystemof interest,we need
to perform a quantum measurement and extract the desired information
from the obtained data. Needless to say, we had better optimize the above
two processes for obtaining information of the quantum system. This re-
searchareaiscalled“QuantumStatisticalInference,”andismuchrequired
for realizing many quantum information processing tasks. This research
field was initiated in the middle of 1960s, and was studied up to 1980 by
American and Russian researchers, for example Holevo, Yuen, Kennedy,
and Belavkin, etc. Their researchis summarized in the following books:
[a] C. W. Helstrom,
Quantum Detection and Estimation Theory,
(Academic Press, New York, 1976).
[b] A. S. Holevo,
Probabilistic and Statistical Aspects of Quantum Theory,
(North-Holland, Amsterdam, 1982); Originally published in Rus-
sian (1980).
However, these researchers did not consider the asymptotic aspects while
the asymptotic theory is essential for the large sample case in statistics.
Moreover, an elegant general theory has been established in classical sta-
tistical inference theory.
Inthemiddleofthe1980s,adifferentresearchdirectionhasbeenstarted
by Nagaoka (who is an expert in mathematical statistics and information
geometry), which focused on the asymptotic theory. In the 1990s, several
Japaneseresearchers(Fujiwara,Matsumoto,Ogawa,Hayashi)havebeenin-
fluenced by Nagaoka, and joined Quantum Statistical Inference. Hence, in
the1990s,bycombiningthemathematicalformulationofquantummechan-
icsandmathematicalstatistics,theseJapaneseresearchersobtainedseveral
good results in Quantum Statistical Inference. Especially, the Japanese
v
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vi Asymptotic Theory of Quantum StatisticalInference
researchers have deeply discussed its asymptotic aspects, which had not
been studied in the earlier stage. Recently, Quantum Statistical Inference
has drawn the attention of several European statisticians (Gill, Bandorff-
Nielsen, Jupp, Gu¸t˘a, Ballester, etc.) who joined this researchfield. On the
otherhand, severaldifferentdirectionsofthis researchareawerestartedin
Europe after the 1990s by physicists, Massar, Popescu, D’Ariano, Buˇzek,
Keyl, Werner, Bagan, Baig, Mun˜oz-Tapia, Gisin, Vidal, Latorre, Pascual,
Tarrachetc. They were motivated by the foundations of physics.
However,despitethisprogress,therehasbeennoEnglishbookcovering
thegreatprogressinquantumstatisticalinferencesinceHolevo’stextbook.
Therefore, this book: Asymptotic Theory of Quantum Statistical Inference
is expected to contribute not only to quantum information science and
statistics but also to various other researchfields.
Moreover,forthereader’sconvenience,Iaddintroductions(Chap0and
introductions for every part), which explain preliminary concepts, histori-
calbackgrounds,andrelatedresearches.Indeed,somecontributorsstrongly
advised me to add introductions for the convienience of readers unfamil-
iar to quantum statistical inference, and agreed with these introductions.
Particularly,acontributorstronglyrecommendedmetointroducesomere-
searchersthatarenotcontainedinthisbook.However,these introductions
are written based only on my personalpoint of view. Thus, I, the editor of
thisbook,assumeallresponsibilityfortheseparts.Theseintroductionsare
different from the collective opinion of all contributors∗. Therefore, they
should be used only as reference materials for the historical significances
of contained papers, and their significances should be decided by readers
themselves.
I wish to thank Japan Science and Technology Agency for covering
the reprint permission fee through ERATO Quantum Computation and
Information Project. I would like to express my gratitude to Professor Hi-
roshi Imai and all members of Quantum Computation and Information
Project for their warm support. I am particularly indebted to Professor
Richard Gill, Professor Keiji Matsumoto, Professor Hiroshi Nagaoka, and
Mr. Fuyuhiko Tanaka for a lot of helpful comments on the introductions.
I am especially grateful to Mr. Fuyuhiko Tanaka, Dr. Kouki Yonezawa,
Mr.SusumuTokumoto,andDr.Yosuke Kikuchiforsupporting the editing
process. Finally, my thanks go to all contributors and translators for their
∗Indeed,acontributorsentmeastrongobjectionagainstcontainingtheseintroductions
becausehedisagreewiththecontent oftheseintroductions.
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Preface vii
contribution and careful proof reading.
Masahito Hayashi
ERATO, Quantum Computation and Information Project
Japan Science and Technology Agency
Hongo, Tokyo, Japan
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viii Asymptotic Theory of Quantum Statistical Inference
December28,2004 13:56 WSPC/Masterfileforreviewvolumewithpartdivider—9inx6in entire
CONTENTS
Preface v
First Appearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0 Introduction to Quantum Statistical Inference . . . . . . . . . . 8
Part I: Hypothesis Testing 17
1 Introduction to Part I . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Strong Converse and Stein’s Lemma in Quantum Hypo-
thesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Tomohiro Ogawa and Hiroshi Nagaoka
3 TheProperFormulaforRelativeEntropyanditsAsymp-
totics in Quantum Probability . . . . . . . . . . . . . . . . . . . 43
Fumio Hiai and D´enes Petz
4 Strong Converse Theorems in Quantum Information Theory . . 64
Hiroshi Nagaoka
5 Asymptotics of Quantum Relative Entropy from a Rep-
resentation Theoretical Viewpoint . . . . . . . . . . . . . . . . . 66
Masahito Hayashi
6 Quantum Birthday Problems: Geometrical Aspects of
Quantum Random Coding . . . . . . . . . . . . . . . . . . . . . 75
Akio Fujiwara
Part II: Quantum Cram´er-Rao Bound in Mixed
States Model 89
7 Introduction to Part II . . . . . . . . . . . . . . . . . . . . . . . 91
ix
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x Asymptotic Theory of Quantum Statistical Inference
8 A New Approach to Cram´er-Rao Bounds for Quantum
State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Hiroshi Nagaoka
9 On Fisher Information of Quantum Statistical Models . . . . . . 113
Hiroshi Nagaoka
10 On the Parameter Estimation Problem for Quantum
Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Hiroshi Nagaoka
11 A Generalization of the Simultaneous Diagonalization
of Hermitian Matrices and its Relation to Quantum
Estimation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 133
Hiroshi Nagaoka
12 A Linear Programming Approach to Attainable Cram´er-
Rao Type Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Masahito Hayashi
13 Statistical Model with Measurement Degree of Freedom
and Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . 162
Masahito Hayashi and Keiji Matsumoto
14 Asymptotic Quantum Theory for the Thermal States Family . . 170
Masahito Hayashi
15 State Estimation for Large Ensembles . . . . . . . . . . . . . . . 178
Richard D. Gill and Serge Massar
Part III: Quantum Cram´er-Rao Bound in Pure
States Model 215
16 Introduction to Part III . . . . . . . . . . . . . . . . . . . . . . . 217
17 Quantum Fisher Metric and Estimation for Pure State Models . 220
Akio Fujiwara and Hiroshi Nagaoka
18 Geometry of Quantum Estimation Theory . . . . . . . . . . . . 229
Akio Fujiwara
19 An Estimation Theoretical Characterizationof Coherent States . 287
Akio Fujiwara and Hiroshi Nagaoka
20 A Geometrical Approach to Quantum Estimation Theory . . . . 305
Keiji Matsumoto
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Contents xi
Part IV: Group Symmetric Approach to Pure
States Model 351
21 Introduction to Part IV . . . . . . . . . . . . . . . . . . . . . . . 353
22 OptimalExtractionofInformationfromFinite Quantum
Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Serge Massar and Sandu Popescu
23 Asymptotic Estimation Theory for a Finite-Dimensional
Pure State Model . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Masahito Hayashi
24 Optimal Universal Quantum Cloning and State Estimation . . . 379
Dagmar Bruß, Artur Ekert, and Chiara Macchiavello
25 Bounds for Generalized Uncertainty of Shift Parameter . . . . . 386
Alexander S. Holevo
Part V: Large Deviation Theory in Quantum Estimation 393
26 Introduction to Part V . . . . . . . . . . . . . . . . . . . . . . . 395
27 OntheRelationbetweenKullbackDivergenceandFisher
Information: From Classical Systems to Quantum Systems . . . 399
Hiroshi Nagaoka
28 Two Quantum Analogues of Fisher Information from a
Large Deviation Viewpoint of Quantum Estimation . . . . . . . 420
Masahito Hayashi
29 Estimating the Spectrum of a Density Operator . . . . . . . . . 458
Michael Keyl and Reinhard F. Werner
Part VI: Further Topics on Quantum Statistical Inference 469
30 Introduction to Part VI . . . . . . . . . . . . . . . . . . . . . . . 471
31 Optimal Quantum Clocks . . . . . . . . . . . . . . . . . . . . . . 477
Vladim´ır Buˇzek, R. Derka, and Serge Massar
32 Quantum Channel Identification Problem . . . . . . . . . . . . . 487
Akio Fujiwara
33 Homodyning as Universal Detection . . . . . . . . . . . . . . . . 494
Giacomo Mauro D’Ariano