Geometrical aspects of entanglement

We study geometrical aspects of entanglement, with the Hilbert-Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a “relativistic” formulation leads to a complete analysis of the question of separability. Our approach is based on Schmidt decomposition of density matrices for a composite system and nonunitary transformations to a standard form. The positivity of the density matrices is crucial for the method to work. A similar approach works to some extent in higher dimensions, but is a less powerful tool. We further present a numerical method for examining separability and illustrate the method by a numerical study of bound entanglement in a composite system of two three-level systems.