General theory of detection and optimality

A general formulation of the problem of detection for a pair of two cones is presented. The special case is the detection of entangled states by entanglement witnesses. Having defined what means "to detect", one can identify the subset of elements, which detect optimally. I will present the properties of this set for a general pair of cones. In particular, I prove the generalization of the theorem of Lewenstein, Krauss, Cirac, Horodecki. The entanglement witness $W$ is optimall if the set of product vectors $\{\phi \otimes \psi: \langle \phi \otimes \psi | W | \phi \otimes \psi \rangle = 0\}$ spans the whole Hilbert space of a system. There exist optimall entangled witness, which do not fullfill this property. It is closely related to some geometrical properties of the boundary of the set of entanglement witnesses and it is possible to say something more about location of such extraordinary states.