Manipulating multi-partite entanglement witnesses using linear programming

An alternative method to detect entanglement is to construct so-called entanglement witnesses (EWs). Entanglement witnesses are physical observables that can detect the presence of entanglement. Recently many attempts have been made to use the convex optimization methods as a robust tool in most quantum information areas. From a different point of view, a very useful approach to construct EWs is the linear programming (LP), a special case of convex optimization. In addition to simplicity, the source of this usefulness comes from the fact that some of the problems can be converted to the (approximate) LP ones. Indeed in most cases determining feasible regions needs to use numerical calculation and consequently the problem is solved approximately. Therefore to figure out the problems which can be easily solved exactly in an optimal way and also can be generalized to an arbitrary number of particles with different Hilbert spaces is the main purpose. To this aim we introduce a new class of parameterized multi-qudit EWs acting on tensor product of arbitrary dimensional Hilbert spaces called reduction type EWs this paper. Furthermore, we have linked the stabilizer theory and the Clifford group operations with structure of new type EWs, so-called stabilizer EWs and we show the computational difficulty in such problems reduced to LP ones. On the other hand we show that the EWs corresponding to the hyperplanes surrounding the feasible regions are optimal in the most cases.