In this thesis, quantum states of two level systems (called qubits) as well as d-level systems (called qudits), d < 1, are investigated. The focus is on characterizing the property of separability and entanglement of composite systems geometrically. Since to some extend this has already been investigated for two qubit or two qudits, this work presents possible generalizations for systems comprising more particles (multipartite). It also shows analytically and numerically how these

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... s affect the prevalent separability criteria such as the Peres-Horodecki criterion, the realignment criterion, the distillation of entanglement and entanglement measures. It is shown in detail how a simplex of n-partite qubit states can be constructed in the similar manner to the bipartite qubit case and why all its elements are bound entangled states. This result is published in Physical Review A 78, 042327 (2008). Furthermore, two different constructions of an n-partite Wk-simplex are presented, which both coincide for n = 2 with the famous magic tetrahedron. For the special cases of the tripartite Wk-simplices (W-state simplices) special symmetries of the eligible quantum systems according to the mentioned separability criteria are revealed and certain subclasses of states can be discriminated. In addition to unveiling these symmetries via this geometrical representation of quantum states, the different cuts of such simplices also allow a precise comparison of the different criteria in a visual and easy way. These facts are therefore contributing to understand and characterize composite quantum systems as well as the associated exciting phenomenon of entanglement and its future applications, such as quantum cryptography, quantum communication or a possible quantum computer.