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A Gramian Approach to Entanglement in Bipartite Finite Dimensional Systems: The case of pure states

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.