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Quantum Physics and Geometry

E-BookPDF1 - PDF WatermarkE-Book
173 Seiten
Englisch
Springer International Publishingerschienen am13.03.20191st ed. 2019
This book collects independent contributions on current developments in quantum information theory, a very interdisciplinary field at the intersection of physics, computer science and mathematics. Making intense use of the most advanced concepts from each discipline, the authors give in each contribution pedagogical introductions to the main concepts underlying their present research and present a personal perspective on some of the most exciting open problems.

Keeping this diverse audience in mind, special efforts have been made to ensure that the basic concepts underlying quantum information are covered in an understandable way for mathematical readers, who can find there new open challenges for their research. At the same time, the volume can also be of use to physicists wishing to learn advanced mathematical tools, especially of differential and algebraic geometric nature.mehr
Verfügbare Formate
BuchKartoniert, Paperback
EUR64,19
E-BookPDF1 - PDF WatermarkE-Book
EUR64,19

Produkt

KlappentextThis book collects independent contributions on current developments in quantum information theory, a very interdisciplinary field at the intersection of physics, computer science and mathematics. Making intense use of the most advanced concepts from each discipline, the authors give in each contribution pedagogical introductions to the main concepts underlying their present research and present a personal perspective on some of the most exciting open problems.

Keeping this diverse audience in mind, special efforts have been made to ensure that the basic concepts underlying quantum information are covered in an understandable way for mathematical readers, who can find there new open challenges for their research. At the same time, the volume can also be of use to physicists wishing to learn advanced mathematical tools, especially of differential and algebraic geometric nature.
Details
Weitere ISBN/GTIN9783030061227
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format Hinweis1 - PDF Watermark
FormatE107
Erscheinungsjahr2019
Erscheinungsdatum13.03.2019
Auflage1st ed. 2019
ReiheLecture Notes of the Unione Matematica Italiana
Reihen-Nr.25
Seiten173 Seiten
SpracheEnglisch
IllustrationenVII, 173 p. 30 illus., 4 illus. in color.
Artikel-Nr.4265926
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Contents;6
2;Contributors;7
3;1 Introduction;8
3.1;References;11
4;2 A Very Brief Introduction to Quantum Computing and Quantum Information Theory for Mathematicians;12
4.1;2.1 Overview;12
4.2;2.2 Quantum Computation as Generalized Probabilistic Computation;13
4.2.1;2.2.1 Classical and Probabilistic Computing via Linear Algebra;13
4.2.2;2.2.2 A Wish List;15
4.2.3;2.2.3 Postulates of Quantum Mechanics and Relevant Linear Algebra;17
4.3;2.3 Entanglement Phenomena;19
4.3.1;2.3.1 Super-Dense Coding;19
4.3.2;2.3.2 Quantum Teleportation;20
4.3.3;2.3.3 Bell's Game;21
4.3.3.1;2.3.3.1 Classical Version;21
4.3.3.2;2.3.3.2 Quantum Version;21
4.4;2.4 Quantum Algorithms;22
4.4.1;2.4.1 Grover's Search Algorithm;22
4.4.2;2.4.2 The Quantum Discrete Fourier Transform;24
4.4.3;2.4.3 The Hidden Subgroup Problem;25
4.5;2.5 Classical Information Theory;26
4.5.1;2.5.1 Data Compression: Noiseless Channels;26
4.5.2;2.5.2 Transmission over Noisy Channels;29
4.5.2.1;2.5.2.1 Capacity of a Noisy Channel;30
4.6;2.6 Reformulation of Quantum Mechanics;30
4.6.1;2.6.1 Partial Measurements;30
4.6.2;2.6.2 Mixing Classical and Quantum Probability;31
4.6.3;2.6.3 Reformulation of the Postulates of Quantum Mechanics;33
4.6.4;2.6.4 Expectation and the Uncertainty Principle;33
4.6.5;2.6.5 Pure and Mixed States;34
4.7;2.7 Communication Across a Quantum Channel;35
4.8;2.8 More on von Neumann Entropy and Its Variants;36
4.9;2.9 Entanglement and LOCC;37
4.9.1;2.9.1 LOCC;38
4.9.2;2.9.2 A Partial Order on Probability Distributions Compatible with Entropy;39
4.9.3;2.9.3 A Reduction Theorem;39
4.9.4;2.9.4 Entanglement Distillation (Concentration) and Dilution;40
4.10;2.10 Tensor Network States;41
4.11;2.11 Representation Theory in Quantum Information Theory;44
4.11.1;2.11.1 Review of Relevant Representation Theory;45
4.11.2;2.11.2 Quantum Marginals and Projections onto Isotypic Subspaces of H d;45
4.12;References;47
5;3 Entanglement, CP-Maps and Quantum Communications;49
5.1;3.1 Introduction;49
5.2;3.2 Entanglement;54
5.2.1;3.2.1 Quantum Correlations and EPR Paradox;54
5.2.2;3.2.2 Sample of Separability Criteria;56
5.3;3.3 Quantum Channels;57
5.3.1;3.3.1 Completely Positive Maps;57
5.3.2;3.3.2 Stinespring Representation;60
5.3.3;3.3.3 Noisy Channels;62
5.4;3.4 Quantum Communications;63
5.4.1;3.4.1 Information Processing;64
5.4.2;3.4.2 Relevant No-Go Theorems: Impossible Machines;65
5.4.3;3.4.3 Quantum Teleportation;68
5.4.4;3.4.4 Dense Coding;72
5.5;3.5 Final Remarks and Perspectives;74
5.6;References;75
6;4 Frontiers of Open Quantum System Dynamics;77
6.1;4.1 Introduction;77
6.2;4.2 Open Quantum System Dynamics;78
6.3;4.3 Characterization of Dynamics with Memory;80
6.3.1;4.3.1 Generalized Non-Markovianity Measure;84
6.4;4.4 Non-Markovian Evolution Equations;85
6.5;4.5 Conclusions and Outlook;89
6.6;References;89
7;5 Geometric Constructions over C and F2 for Quantum Information;92
7.1;5.1 Introduction;92
7.2;5.2 The Geometry of Entanglement;96
7.2.1;5.2.1 Entanglement Under SLOCC, Tensor Rank and Algebraic Geometry;96
7.2.2;5.2.2 The Three-Qubit Classification via Auxiliary Varieties;101
7.2.3;5.2.3 Geometry of Hyperplanes: The Dual Variety;102
7.2.4;5.2.4 Representation Theory and Quantum Systems;106
7.2.5;5.2.5 From Sequence of Simple Lie Algebras to the Classification of Tripartite Quantum Systems with Similar Classes of Entanglement;109
7.3;5.3 The Geometry of Contextuality;113
7.3.1;5.3.1 Observable-Based Proofs of Contextuality;113
7.3.2;5.3.2 The Symplectic Polar Space of Rank N and the N-Qubit Pauli Group;114
7.3.3;5.3.3 Geometry of Hyperplanes: Veldkamp Space of a Point-Line Geometry;116
7.3.4;5.3.4 The Finite Geometry of the Two-Qubit and Three-Qubit Pauli Groups and the Hyperplanes of W(2N-1,2);117
7.3.5;5.3.5 From Commutation Relations of the Three-Qubit Pauli Group to the Weight Diagrams of Simple Lie Algebras;122
7.4;5.4 Conclusion;125
7.5;References;126
8;6 Hilbert Functions and Tensor Analysis;130
8.1;6.1 Introduction;130
8.2;6.2 Tensors and Projective Geometry;131
8.2.1;6.2.1 The Hilbert Function of Finite Sets in Projective Spaces;134
8.3;6.3 Results on Tensors from Classical Projective Geometry;141
8.4;6.4 Kruskal's Criterion and Terracini's Criterion;146
8.5;6.5 A New Result on the Decomposition of Tensors;150
8.6;6.6 Final Remarks and Open Problems;153
8.7;References;155
9;7 Differential Geometry of Quantum States, Observablesand Evolution;157
9.1;7.1 Introduction;157
9.1.1;7.1.1 On the Many Pictures of Quantum Mechanics;158
9.1.2;7.1.2 Dirac-Schrödinger vs. Heisenberg-Weyl Picture;158
9.1.2.1;7.1.2.1 Dirac-Schrödinger Picture;159
9.1.2.2;7.1.2.2 Heisenberg-Born-Jordan;160
9.1.2.3;7.1.2.3 Other Pictures;160
9.2;7.2 A Geometric Picture of Quantum Mechanics;161
9.2.1;7.2.1 Quantum States and Open Systems;169
9.2.1.1;7.2.1.1 The Qubit;170
9.2.1.2;7.2.1.2 Open Quantum Systems: the GKLS Equation;172
9.3;7.3 Composition of Systems;173
9.3.1;7.3.1 Decomposing a System;173
9.4;7.4 Conclusions and Discussion;174
9.5;References;175mehr