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We introduce with geometric means a density matrix decomposition of amultipartite quantum system of a finite dimension into two density matrices: aseparable one, also known as the best separable approximation, and anessentially entangled one, which contains no product states components. We showthat this convex decomposition can be achieved in practice with the help of analgorithm based on linear programming, which in the general case scalespolynomially with the dimension of the multipartite system. Furthermore, wesuggest methods for analyzing the multipartite entanglement content of theessentially entangled component and derive analytically an upper bound for itsrank. We illustrate the algorithm at an example of a composed system of totaldimension 12 undergoing loss of coherence due to classical noise and we tracethe time evolution of its essentially entangled component. We suggest a"geometric" description of entanglement dynamics and show how it explains thewell-known phenomena of sudden death and revival of multipartite entanglement.

The Essentially Entangled Component of Multipartite Mixed Quantum States, its Properties and an Efficient Algorithm for its Extraction