Asymptotic theory of quantum statistical inference

Masahito Hayashi9812560157, 9789812560155, 9789812563071

Quantum statistical inference, a research field with deep roots in the foundations of both quantum physics and mathematical statistics, has made remarkable progress since 1990. In particular, its asymptotic theory has been developed during this period. However, there has hitherto been no book covering this remarkable progress after 1990; the famous textbooks by Holevo and Helstrom deal only with research results in the earlier stage (1960s-1970s). This book presents the important and recent results of quantum statistical inference. It focuses on the asymptotic theory, which is one of the central issues of mathematical statistics and had not been investigated in quantum statistical inference until the early 1980s. It contains outstanding papers after Holevo’s textbook, some of which are of great importance but are not available now. The reader is expected to have only elementary mathematical knowledge, and therefore much of the content will be accessible to graduate students as well as research workers in related fields. Introductions to quantum statistical inference have been specially written for the book. Asymptotic Theory of Quantum Statistical Inference: Selected Papers will give the reader a new insight into physics and statistical inference.

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Table of contents :
Preface……Page 4
Contents……Page 8
First Appearance……Page 12
List of Contributors……Page 16
Introduction to Quantum Statistical Inference……Page 19
Introduction to Part I……Page 30
Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing……Page 39
The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability……Page 54
Strong Converse Theorems in Quantum Information Theory……Page 75
Asymptotics of Quantum Relative Entropy from a Representation Theoretical Viewpoint……Page 77
Quantum Birthday Problems: Geometrical Aspects of Quantum Random Coding……Page 86
Part II – Quantum Cram´er-Rao Bound in Mixed States Model……Page 100
Introduction to Part II……Page 102
A New Approach to Cram´er-Rao Bounds for Quantum State Estimation……Page 111
On Fisher Information of Quantum Statistical Models……Page 124
On the Parameter Estimation Problem for Quantum Statistical Models……Page 136
A Generalization of the Simultaneous Diagonalization of Hermitian Matrices and its Relation to Quantum Estimation Theory……Page 144
A Linear Programming Approach to Attainable Cram´er-Rao Type Bounds……Page 161
Statistical Model with Measurement Degree of Freedom and Quantum Physics……Page 173
Asymptotic Quantum Theory for the Thermal States Family……Page 181
State Estimation for Large Ensembles……Page 189
Part III – Quantum Cram´er-Rao Bound in Pure States Model……Page 226
Introduction to Part III……Page 228
Quantum Fisher Metric and Estimation for Pure State Models……Page 231
Geometry of Quantum Estimation Theory……Page 240
An Estimation Theoretical Characterization of Coherent States……Page 298
A Geometrical Approach to Quantum Estimation Theory……Page 316
Part IV – Group Symmetric Approach to Pure States Model……Page 362
Introduction to Part IV……Page 364
Asymptotic Estimation Theory for a Finite-Dimensional Pure State Model……Page 376
Optimal Universal Quantum Cloning and State Estimation……Page 390
Bounds for Generalized Uncertainty of Shift Parameter……Page 397
Optimal Extraction of Information from Finite Quantum Ensembles……Page 367
Part V – Large Deviation Theory in Quantum Estimation……Page 404
Introduction to Part V……Page 406
On the Relation between Kullback Divergence and Fisher Information: From Classical Systems to Quantum Systems……Page 410
Two Quantum Analogues of Fisher Information from a Large Deviation Viewpoint of Quantum Estimation……Page 431
Estimating the Spectrum of a Density Operator……Page 469
Part IV – Further Topics on Quantum Statistical Inference……Page 480
Introduction to Part VI……Page 482
Optimal Quantum Clocks……Page 488
Quantum Channel Identification Problem……Page 498
Homodyning as Universal Detection……Page 505
On the Measurement of Qubits……Page 520
Index……Page 550