Huber, Felix Michael: Quantum states and their marginals : from multipartite entanglement to quantum error-correcting codesQuantenzustände und deren Marginalien : von multipartiter Verschränkung zu fehlerkorrigierenden Quantencodes. 2017
Inhalt
- Title page
- Abstract
- Zusammenfassung
- Contents
- List of Illustrations
- Preface
- Chapter 1. Basic concepts
- 1.1 Fundamentals
- 1.1.1 Quantum states
- 1.1.2 Multiple particles
- 1.1.3 Operators and maps
- 1.1.4 Entanglement
- 1.1.5 Entropy and distance measures
- 1.1.6 Entanglement detection and measures
- 1.1.7 Bloch representation
- 1.2 Further notions
- Chapter 2. Ground and thermal states of local Hamiltonians
- 2.1 Introduction
- 2.2 The setting
- 2.3 Characterization of conv(Qk) via semidefinite programming
- 2.4 Characterization via the graph state formalism
- 2.5 Quantum simulation as an application
- 2.6 Further results
- 2.6.1 Further results on the information projection
- 2.6.2 Ground and excited states of local Hamiltonians
- 2.6.3 States of four parties
- 2.6.4 Even- and odd-body correlations of qubit states
- 2.7 Conclusion
- Chapter 3. AME state of seven qubits
- 3.1 The Bloch representation
- 3.2 Properties of AME state reductions
- 3.3 Scott bound
- 3.4 Nonexistence of the seven qubit AME state
- 3.5 Upper bound for the number of maximally mixed reductions
- 3.6 AME states of n qubits
- 3.7 Further results
- 3.8 Conclusion
- Chapter 4. Ulam's problems for quantum states
- 4.1 Motivation
- 4.2 Realizability and uniqueness in graphs
- 4.3 Graph states
- 4.4 Weight distribution
- 4.5 Constraints on the weight distribution
- 4.6 Detecting illegitimate decks
- 4.7 When is a weight distribution graphical?
- 4.8 Conclusion
- Chapter 5. Constraints on QECC and AME states
- 5.1 Introduction
- 5.2 Motivation
- 5.3 The Bloch representation
- 5.4 Quantum error-correcting codes
- 5.5 The shadow enumerator
- 5.6 Shor-Laflamme enumerators
- 5.7 The quantum MacWilliams identity
- 5.8 The shadow enumerator in terms of the Shor-Laflamme enumerator
- 5.9 New bounds on absolutely maximally entangled states
- 5.10 Discussion
- 5.11 Further results
- 5.11.1 A generalization of the universal state inversion from the shadow inequalities
- 5.11.2 An application to the quantum marginal problem
- 5.11.3 A strong subadditivity - like expression for the linear entropy
- 5.11.4 Further non-existence results of qubit-codes
- 5.11.4 Weight distributions of hypothetical codes
- 5.12 Conclusion
- Summary and outlook
- Acknowledgments
- Appendix: Krawtchouk polynomials
- Bibliography