Twopartite, combinatorial approach to partial k-separability problem for general multipartite states

AbstractWe describe a general methods to localize any sort of k-separability and therefore also the correspondingpartial entanglement in genuinely multipartite mixed quantum states. Our methods are based exclusivelyon the known twopartite methods and some systematic procedures of combinatorial nature. Our methodsare formalized in an algorithmic-like fashion and therefore they are easily implementable in a computerenvironments and might be eﬀectively used for studying numerically these questions for suﬃciently smallsystems 1 Introduction Quantum entanglement, since its discovery by Einstein, Podolsky and Rosen [1], and Schrodinger [2] at-tracted very large attention, especially in the past two decades due to its intriguing properties and extremelyimportant signiﬁcance in the quantum information processing tasks [3], [4], [11]. The main eﬀorts have beenfocused on bipartite entanglement and at present many interesting and deep results have been obtained inthis case, see [5], [6] for some recent reviews.The present note is addressed to a very preliminary discussion of certain aspects of multipartite entan-glement. The importance of multipartite entanglement can be illustrated by several illuminating examples.A class of the so called graph states used in one-way quantum computation model [7] and fault-toleranttopological quantum computation [21] model as well are both good examples. Multipartite entangled pho-tons are essential resources used in quantum key distribution protocols [8], [9]. For the purposes of quantumcommunication a multipartite entangled states can serve as quantum communication channels in most of theknown teleportation protocols [10]. Multipartite entanglement displays one of the most fascinating featuresof quantum theory that is called as nonlocality of the quantum world [13]. Comparing to the best known two-partite case [4], [5], [6], the situation concerning a general multipartitesystem is much more complicated even on the level of classiﬁcation of the possible entanglements that canarise in such systems. Given a multipartite state, a natural question to ask is whether it is entangled stateand if so how to measure the amount of entanglement included.The main aim of the present note is to demonstrate how one can use exclusively two-partite systemsmethods to answer the ﬁrst question for a general multipartite quantum state, see also [18], [19] for somerecent references on this topics. The presented in this note methods are being implemented and applied tostudy several particular (low dimensional mainly) systems within constructed in Zielona Gora University inthe past few years simulator of quantum computation machine, called Quantum Computing System (QCS)which is presented partially in [20] and will be presented in [14] with all details necessary for this. Thereport from our computer experiments will be presented elsewhere.The present note is organized in the following way. In the next section basic deﬁnitions will be introducedand the algorithm detecting and localizing entanglements in the case of pure states will be formulated. Anextension of this method to general multipartite mixed states will be presented in Section (3).