ua nt - p h / 02 07 05 8 v 3 2 2 O ct 2 00 2 Combinatorial topology of multipartite entangled states

With any state of a multipartite quantum system its separability polytope is associated. This is an algebro-topological object (nontrivial only for mixed states) which captures the localisation of entanglement of the state. Particular examples of separability polytopes for 3-partite systems are explicitly provided. It turns out that this characterisation of entanglement is associated with simulation of arbitrary unitary operations by 1and 2-qubit gates. A topological description of how entanglement changes in course of such simulation is provided. Introduction Entanglement in multipartite quantum system is now treated as a key resource in quantum information processing. That is why multiple efforts are drawn to quantification of entanglement for quantum states. For bipartite quantum systems all entanglement measures are essentially of numerical nature as a single real positive number is enough to quantify the degree of entanglement. In the case of multipartite systems the situation differs drastically. It was observed that even pure states of a 3-particle system can be entangled in different ways, which can not be interconverted by local unitary transformations.