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E-grāmata: Introduction to Quantum Information Science

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  • Formāts: PDF+DRM
  • Sērija : Graduate Texts in Physics
  • Izdošanas datums: 22-Aug-2014
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783662435021
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Introduction to Quantum Information SciencePalielināt
  • Formāts: PDF+DRM
  • Sērija : Graduate Texts in Physics
  • Izdošanas datums: 22-Aug-2014
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783662435021
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This book presents the basics of quantum information, e.g., foundation of quantum theory, quantum algorithms, quantum entanglement, quantum entropies, quantum coding, quantum error correction and quantum cryptography. The required knowledge is only elementary calculus and linear algebra. This way the book can be understood by undergraduate students. In order to study quantum information, one usually has to study the foundation of quantum theory. This book describes it from more an operational viewpoint which is suitable for quantum information while traditional textbooks of quantum theory lack this viewpoint. The current book bases on Shor's algorithm, Grover's algorithm, Deutsch-Jozsa's algorithm as basic algorithms. To treat several topics in quantum information, this book covers several kinds of information quantities in quantum systems including von Neumann entropy. The limits of several kinds of quantum information processing are given. As important quantum protocols, this book contains quantum teleportation, quantum dense coding, quantum data compression. In particular conversion theory of entanglement via local operation and classical communication are treated too. This theory provides the quantification of entanglement, which coincides with von Neumann entropy. The next part treats the quantum hypothesis testing. The decision problem of two candidates of the unknown state are given. The asymptotic performance of this problem is characterized by information quantities. Using this result, the optimal performance of classical information transmission via noisy quantum channel is derived. Quantum information transmission via noisy quantum channel by quantum error correction are discussed too. Based on this topic, the secure quantum communication is explained. In particular, the quantification of quantum security which has not been treated in existing book is explained. This book treats quantum cryptography from a more practical viewpoint.
1 Invitation to Quantum Information Science
1(12)
1.1 From Classical Information Science to Quantum Information Science
1(3)
1.2 Further Expansion of Quantum Information Science
4(2)
1.3 Feedback from Quantum Information Science to Physics
6(2)
1.4 Toward Realization of Quantum Information Processing
8(1)
1.5 Organization of This Book
9(4)
References
12(1)
2 Quantum Mechanics for Qubit Systems
13(24)
2.1 Preliminary
13(2)
2.2 Preparation
15(5)
2.2.1 Conceptual Preparation: Physical System, State, Measurement of Physical Quantity
15(1)
2.2.2 Notational Preparation: Dirac Notation
16(4)
2.3 Qubit Systems
20(17)
2.3.1 A Qubit System
21(3)
2.3.2 Time Evolution in Qubit System
24(4)
2.3.3 Composition of Qubit Systems: n-Qubit Systems
28(6)
References
34(3)
3 Foundations on Quantum Computing
37(18)
3.1 What is Computation?
37(3)
3.2 Mathematical Notation for Information Science
40(1)
3.3 Classical Circuit Model
41(4)
3.4 Quantum Circuit Model
45(10)
References
53(2)
4 Quantum Algorithms
55(20)
4.1 Introduction
55(1)
4.2 Deutsch-Jozsa Algorithm
56(3)
4.3 Grover's Algorithm
59(7)
4.3.1 Construction of Grover's Algorithm
59(3)
4.3.2 Analysis of Success Probability
62(2)
4.3.3 Generalization: Multiple Solutions
64(2)
4.4 Shor's Algorithm
66(7)
4.4.1 Quantum Algorithm for Period Finding
66(4)
4.4.2 Quantum Algorithm for Factorization
70(2)
4.4.3 Quantum Algorithm for Discrete Logarithm
72(1)
4.5 Other Quantum Algorithms
73(2)
References
73(2)
5 Foundations of Quantum Mechanics and Quantum Information Theory
75(52)
5.1 Introduction
75(5)
5.1.1 Postulates and Preconditions
76(1)
5.1.2 Hilbert Space and Linear Operators
77(1)
5.1.3 Dirac Notation II
78(2)
5.2 Postulates for Quantum Mechanics
80(13)
5.2.1 Quantum States, and Measurements of Physical Quantities
81(7)
5.2.2 Time Evolution
88(1)
5.2.3 Composite Systems
89(2)
5.2.4 Comment on the Measurement Process: State-Changes due to Measurements
91(2)
5.3 Reformulation of Quantum Mechanics
93(34)
5.3.1 General Class of Quantum States
93(14)
5.3.2 General Class of Measurements
107(8)
5.3.3 General Class of Time Evolutions
115(4)
5.3.4 General Class of Measurement Processes
119(6)
References
125(2)
6 Information Quantities in Quantum Systems
127(40)
6.1 Introduction
127(1)
6.2 Information Quantities in Classical Systems
128(17)
6.2.1 Shannon Entropy
128(2)
6.2.2 Entropy and Typical Sequence
130(4)
6.2.3 Joint Entropy and Conditional Entropy
134(1)
6.2.4 Conditional Probability and Classical Channel
135(1)
6.2.5 Divergence
136(2)
6.2.6 f-Divergence
138(3)
6.2.7 Mutual Information
141(2)
6.2.8 Concavity and Subadditivity of the Entropy
143(1)
6.2.9 Fano Inequality
144(1)
6.3 Information Quantities in Quantum Systems
145(22)
6.3.1 Von Neumann Entropy
145(2)
6.3.2 Quantum Relative Entropy
147(2)
6.3.3 Mutual Information in Quantum Systems
149(3)
6.3.4 Concavity and Subadditivity of the von Neumann Entropy
152(1)
6.3.5 Trace Distance
153(2)
6.3.6 Fidelity and Uhlmann's Theorem
155(3)
6.3.7 Properties of Fidelity
158(4)
6.3.8 Entanglement Fidelity
162(3)
References
165(2)
7 Quantum Entanglement
167(38)
7.1 Introduction
167(1)
7.2 Basic Concepts of Entanglement
168(6)
7.2.1 Quantum and Classical Correlation
168(1)
7.2.2 Product State and Maximally Entangled State
169(2)
7.2.3 Quantum Teleportation
171(2)
7.2.4 Superdense Coding
173(1)
7.3 Quantifying Entanglement
174(12)
7.3.1 Local Operations and Classical Communication (LOCC)
174(2)
7.3.2 Basic Unit of Entanglement
176(1)
7.3.3 Entanglement Concentration
177(3)
7.3.4 Quantum Data Compression
180(3)
7.3.5 Entanglement Dilution
183(2)
7.3.6 Amount of Entanglement
185(1)
7.4 Multipartite Entanglement
186(4)
7.4.1 GHZ and W State
186(1)
7.4.2 Stochastic LOCC
187(1)
7.4.3 Classification of Multipartite Entanglement
188(2)
7.5 Mixed-State Entanglement
190(15)
7.5.1 Entanglement Criteria
190(4)
7.5.2 LOCC on Mixed States
194(2)
7.5.3 Entanglement Measure
196(1)
7.5.4 Entanglement of Formation
197(2)
7.5.5 Relative Entropy of Entanglement
199(1)
7.5.6 Relationship among Entanglement Measures
200(2)
7.5.7 Entanglement Monotone
202(1)
References
202(3)
8 Classical-Quantum Channel Coding
205(26)
8.1 Introduction
205(1)
8.2 Quantum Hypothesis Testing
206(10)
8.2.1 Problem of Quantum Hypothesis Testing
206(2)
8.2.2 Trace Inequality for Quantum Hypothesis Testing
208(2)
8.2.3 Asymptotic Theory of Quantum Hypothesis Testing
210(3)
8.2.4 Properties of Relative Renyi Entropy
213(3)
8.3 Classical-Quantum Channel Coding
216(15)
8.3.1 Message Transmission Over Quantum Channels
216(3)
8.3.2 Proof of the C-Q Channel Coding Theorem (Converse Part)
219(4)
8.3.3 Proof of the C-Q Channel Coding Theorem (Direct Part)
223(5)
References
228(3)
9 Quantum Error Correction and Quantum Cryptography
231(38)
9.1 Introduction
231(1)
9.2 Algebraic Error Correction in the Classical System
232(14)
9.2.1 Formulation
232(4)
9.2.2 Evaluation of Error Probability Under Code Ensemble
236(3)
9.2.3 Examples of Code Ensemble
239(2)
9.2.4 Asymptotic Theory
241(1)
9.2.5 Error Correction with Confidentiality
242(4)
9.3 Quantum Error Correcting Code
246(9)
9.3.1 Pauli Channel
246(1)
9.3.2 Stabilizer Code
247(3)
9.3.3 CSS Code
250(4)
9.3.4 Asymptotic Theory
254(1)
9.4 Application to Quantum Secret Communication
255(9)
9.4.1 Channel to the Environment System
255(1)
9.4.2 Leaked Information Without Privacy Amplification
256(4)
9.4.3 Leaked Information with Privacy Amplification
260(4)
9.5 Application to Quantum Cryptography (Quantum Key Distribution)
264(5)
References
268(1)
Appendix A Foundations of Linear Algebra and Basic Mathematics 269(36)
Appendix B Solution for Exercises 305(22)
Index 327
Masahito Hayashi has received his Ph.D. at Kyoto University, Department of Mathematics, Graduate School of Science, in 1999. Between 1998 and 2000 he was a Research Fellow of the Japan Society for the Promotion of Science followed by three years of research at the Laboratory for Mathematical Neuroscience (Amari's Laboratory), Brain Science Institute, RIKEN. From 2003 to 2006 he held the position of a Research Manager at ERATO, Quantum Computation and Information Project, Japan Science and Technology Agency (JST). During that period he oversaw the ERATO Quantum Computation and Information Project, which at the time was the largest research group for quantum information and computation in Japan. Afterwards he became a Group Leader, Quantum Information Theory Group, ERATO-SORST Quantum Computation and Information Project. In 2007 he joined Tohoku University as Associate Professor at the Graduate School of Information Sciences, Tohoku University. In 2012 he became a Full Professor of Nagoya University, Graduate School of Mathematics.

Masahito Hayashi is one of the founders of the Asian Conference on Quantum Information Science (AQIS) conference series, which is a major international conference series on quantum information and computation. He has served as a referee for numerous prestigious international journals.
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