Intro; Contents; Contributors; 1 Introduction; References; 2 A Very Brief Introduction to Quantum Computing and Quantum Information Theory for Mathematicians; 2.1 Overview; 2.2 Quantum Computation as Generalized Probabilistic Computation; 2.2.1 Classical and Probabilistic Computing via Linear Algebra; 2.2.2 A Wish List; 2.2.3 Postulates of Quantum Mechanics and Relevant Linear Algebra; 2.3 Entanglement Phenomena; 2.3.1 Super-Dense Coding; 2.3.2 Quantum Teleportation; 2.3.3 Bell's Game; 2.3.3.1 Classical Version; 2.3.3.2 Quantum Version; 2.4 Quantum Algorithms; 2.4.1 Grover's Search Algorithm

2.4.2 The Quantum Discrete Fourier Transform2.4.3 The Hidden Subgroup Problem; 2.5 Classical Information Theory; 2.5.1 Data Compression: Noiseless Channels; 2.5.2 Transmission over Noisy Channels; 2.5.2.1 Capacity of a Noisy Channel; 2.6 Reformulation of Quantum Mechanics; 2.6.1 Partial Measurements; 2.6.2 Mixing Classical and Quantum Probability; 2.6.3 Reformulation of the Postulates of Quantum Mechanics; 2.6.4 Expectation and the Uncertainty Principle; 2.6.5 Pure and Mixed States; 2.7 Communication Across a Quantum Channel; 2.8 More on von Neumann Entropy and Its Variants

2.9 Entanglement and LOCC2.9.1 LOCC; 2.9.2 A Partial Order on Probability Distributions Compatible with Entropy; 2.9.3 A Reduction Theorem; 2.9.4 Entanglement Distillation (Concentration) and Dilution; 2.10 Tensor Network States; 2.11 Representation Theory in Quantum Information Theory; 2.11.1 Review of Relevant Representation Theory; 2.11.2 Quantum Marginals and Projections onto Isotypic Subspaces of H d; References; 3 Entanglement, CP-Maps and Quantum Communications; 3.1 Introduction; 3.2 Entanglement; 3.2.1 Quantum Correlations and EPR Paradox; 3.2.2 Sample of Separability Criteria

3.3 Quantum Channels3.3.1 Completely Positive Maps; 3.3.2 Stinespring Representation; 3.3.3 Noisy Channels; 3.4 Quantum Communications; 3.4.1 Information Processing; 3.4.2 Relevant No-Go Theorems: Impossible Machines; 3.4.3 Quantum Teleportation; 3.4.4 Dense Coding; 3.5 Final Remarks and Perspectives; References; 4 Frontiers of Open Quantum System Dynamics; 4.1 Introduction; 4.2 Open Quantum System Dynamics; 4.3 Characterization of Dynamics with Memory; 4.3.1 Generalized Non-Markovianity Measure; 4.4 Non-Markovian Evolution Equations; 4.5 Conclusions and Outlook; References

5 Geometric Constructions over C and F2 for Quantum Information5.1 Introduction; 5.2 The Geometry of Entanglement; 5.2.1 Entanglement Under SLOCC, Tensor Rank and Algebraic Geometry; 5.2.2 The Three-Qubit Classification via Auxiliary Varieties; 5.2.3 Geometry of Hyperplanes: The Dual Variety; 5.2.4 Representation Theory and Quantum Systems; 5.2.5 From Sequence of Simple Lie Algebras to the Classification of Tripartite Quantum Systems with Similar Classes of Entanglement; 5.3 The Geometry of Contextuality; 5.3.1 Observable-Based Proofs of Contextuality