Characterizing the structure of multiparticle entanglement in high-dimensional systems

Quantum entanglement is a useful resource for many quantum informational tasks. In this context, enlarging the number of participating systems as well as increasing the system dimension has proven to enhance the performance. In order to successfully use this resource, it is crucial to have a consistent theoretical description of the different kinds of entanglement that can occur within those systems.
This thesis studies the classification of entanglement in special families of multipartite and higher dimensional quantum systems. Furthermore, attention is put to the detection of entanglement within these systems. There are three main projects addressed within this thesis. The first is concerned with the detection of entanglement between multiple systems based on the construction of entanglement witnesses. Here, a one-to-one connection between SLOCC-witnesses and entanglement witnesses within an enlarged Hilbert space is made. The form of the witness operator is such that it can be constructed from any representative state of the corresponding SLOCC class and its maximal overlap with the set of separable states or the set of states within another SLOCC class.
Within the second part, a special family of multipartite quantum states, the so-called qubit hypergraph states, is generalized to arbitrary dimensions. Following the definition of the basic framework, relaying strongly on the phase-space description of quantum states, rules to categorize qudit hypergraph states with respect to SLOCC- as well as LU-equivalence are determined. Interestingly, there exist close connections to the field of number theory. Furthermore, a full classification in terms of SLOCC and LU is provided for tripartite systems of dimension three and four. Within the subsequent section, rules for local complementation within graph states of not-neccssarily prime dimension are presented. Finally, an extension to weighted hypergraphs is made and, for some particular cases, SLOCC equivalence classes are determined.
The third and last part of this thesis is dedicated to the question of how to reasonably define genuine multilevel entanglement. Starting from an example, a discrepancy of the widely used term of a maximally entangled state and the practical resources needed to produce such a state is shown. This motivates a definition of genuine multilevel entanglement that adapts to the fact that genuine d-level entangled states should need at least d-dimensional resource states. Based on this, the set of entangled multilevel states is then divided into three classes: decomposable (DEC-) states that can be generated from lower dimensional systems, genuine multilevel, multipartite entangled (GMME-) states, whose correlations cannot be reproduced by lower dimensional systems and multilevel, multipartite entangled (MME-) states which lie in between. That is, the last class covers the set of states, which are decomposable with respect to some bipartition. Naturally, within the bipartite scenario, the set of MME-states coincides with the set of decomposable states. Having set the framework, examples for all three classes are provided, as well as methods to distinguish between those. In the bipartite scenario, an analytical criterion is presented that additionally can be used to differ MME from GMME in the multipartite case. To distinguish MME-states from DEC- states has proven to be more involved, nonetheless there exist successful numerical optimization protocols as well as an necessary but not sufficient analytical criterion.