Testing multipartite productness is easier than testing bipartite productness

We prove a lower bound on the number of copies needed to test the property of a multipartite quantum state being product across some bipartition (i. e. not genuinely multipartite entangled), given the promise that the input state either has this property or is -far in trace distance from any state with this property. We show that (n / n) copies are required (for fixed 12), complementing a previous result that O (n / ²) copies are sufficient. Our proof technique proceeds by considering uniformly random ensembles over such states, and showing that the trace distance between these ensembles becomes arbitrarily small for sufficiently large n unless the number of copies is at least (n / n). We discuss implications for testing graph states and computing the generalised geometric measure of entanglement.