Using non-positive maps to characterize entanglement witnesses

In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach crucial role play such maps which are surjective between sets $\mathcal{P}_{k}^d$ of $k \leq d$ rank projectors and the set $\mathcal{P}_1^d$ of rank one projectors acting in the $d$ dimensional space. We argue that our method can be used to check whether a given observable is an entanglement witness. In the second part of this paper we show that inverse reduction map satisfies this requirement and using it we can obtain a bunch of new entanglement witnesses.