Construction of multi-qubit optimal genuine entanglement witnesses

We interpret multi-partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them. We apply this result to multi-qubit X-shaped Hermitian matrices, and characterize the conditions for them to be genuine entanglement witnesses, in terms of entries. Furthermore, we find all optimal entries among them. They turn out to have spanning properties, and so they detect a non-zero volume set of multi-qubit genuine entanglement. We also characterize decomposability for X-shaped entanglement witnesses.