Inhalt

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   Abstract
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   Zusammenfassung
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   Contents
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   List of Figures
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   List of Tables
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   1 Introduction
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  2 Preliminaries
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  2.1 Mathematical framework
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   2.1.1 Hilbert space
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  2.1.2 Quantum states in Hilbert space
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   Pure quantum states
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   The coefficient matrix of pure quantum states
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   Schmidt decomposition
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   Mixed quantum states
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   Reduced quantum states
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   2.1.3 Quantum operations
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   Global and local quantum operations on multipartite systems
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   Measurements
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   Local Unitary operations
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   LOCC-Operations
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   SLOCC-operations
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  2.2 Entanglement
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   2.2.1 Bipartite entanglement
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   2.2.2 Multipartite entanglement
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   Pure states
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   Mixed states
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  2.2.3 Applications of entanglement
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   Quantum key distribution
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   Quantum metrology
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  2.3 Entanglement classification and quantification
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   2.3.1 Classification
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   2.3.2 Quantification
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   2.3.3 Classification of bipartite entanglement
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   LU and LOCC classification of bipartite qubit states
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   2.3.4 Classification of multipartite entanglement
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  2.4 Entanglement detection
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   2.4.1 PnCP-maps
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   PPT-criterion
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   Reduction criterion
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   Majorization criterion
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   Range criterion
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   Matrix realignement criterion
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   2.4.2 Entanglement witnesses
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   Construction of entanglement witnesses
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   SLOCC witnesses
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  2.5 Graph states, Hypergraph states and the Stabilizer formalism
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   2.5.1 Graph states
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   Graph states as stabilizer states
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   Local complementation of qubit graph states
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   2.5.2 Hypergraph states
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   2.5.3 One way quantum computer - a graph state application
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  3 Tensor witness
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   3.1 SDP - introduction
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  3.2 Tensor witness
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   3.2.1 Introduction
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  3.2.2 Preliminaries
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   SLOCC classes
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   Entanglement witnesses
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   SLOCC witnesses
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   3.2.3 One-to-one correspondence between SLOCC- and entanglement witness
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   PPT-relaxation
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   3.2.4 Numerical values for 2 x 3 x 3
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   3.2.5 Conclusions
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  3.2.6 Appendix
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   Example
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  4 Hypergraph states in arbitrary, finite dimension
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   4.1 Phase space representation of quantum systems in finite Hilbert spaces
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  4.2 Qudit hypergraph states
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   4.2.1 Introduction
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  4.2.2 Background and basic definitions
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   The Pauli group and its normalizer
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   Qudit graph states
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   4.2.3 Qudit hypergraph states
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  4.2.4 Properties of hypergraph states and the stabilizer formalism
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   Local action of Pauli and Clifford groups
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   Stabilizer formalism
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   Local measurements in Z basis and ranks of the reduced states
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  4.2.5 SLOCC and LU classes of hypergraphs
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   SLOCC and LU transformations
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   Tools for SLOCC classification
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   Tools for LU-classification
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  4.2.6 Classification of qudit hypergraphs under SLOCC and LU
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   Elementary hypergraphs
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   Hypergraphs LU-equivalent to elementary hypergraphs
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   4.2.7 Classification of 3 x 3 x 3
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  4.2.8 Classification of 4 x 4 x 4
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   Class 1
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   Class 1'
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   Class 2
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   Class 3
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   Class 4
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   SLOCC-inequivalence of Classes 1-4,1'
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   4.2.9 Conclusions
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  4.2.10 Appendix
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   Phase-space picture
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   Local complementation of qudit graphs in prime dimension
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   Proofs of Proposition 4.1
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   4.3 Local complementation of qudit graph states in arbitrary dimension
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   4.4 Weighted hypergraphs
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   4.4.1 Elementary weighted hypergraphs
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  4.4.2 SLOCC equivalence of weighted hypergraphs
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   SLOCC equivalence of elementary weighted hypergraphs with different weights
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   SLOCC-equivalence via basis mapping
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  5 Characterizing genuine multilevel entanglement
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   5.1 Genuine multilevel entanglement
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   5.1.1 Introduction
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   5.1.2 The scenario
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   5.1.3 General theory for bipartite systems
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   5.1.4 Multiparticle systems
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   5.1.5 Conclusion
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  5.1.6 Appendix
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   A: Proof of Observation 1
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   B: Witnesses for the bipartite case
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   C: Connection to the theory of Young tableaux
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   D: Examples
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   Example 1. GHZ States
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   Example 2: Graph states
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   The maximally entangled state of six qubits
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   E: Algorithm for testing full decomposability
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   5.2 Distinguishing MME from DEC
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   5.3 Lower dimensional representation of qudit graph states
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  5.4 Network configuration
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   5.4.1 Triangular network configuration
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   Examples: GHZ-states and the network configuration
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   6 Summary and Outlook
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   Acknowledgements
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   Appendix A - SLOCC classification of 233 systems
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   Bibliography