Description
When a q-partite state can not be written as a tensor product of less than q-partite entangled state then it is said to have genuine q-partite entanglement. I will consider a special class of local unitary invariants - called multi-invariants - that is multiplicative under tensor factorization. I'll show how one can take a linear combination of their logarithms to build a combination that vanishes for any state that is not genuine q-partite entangled. In this way, the non-vanishing value of this linear combination signals presence of genuinely q-partite entanglement.
