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 $\epsilon$-far in trace distance from any state with this property. We show that $\mathrm{\Omega }(n\mathrm{/}\log n)$ copies are required (for fixed $\epsilon \le \frac{1}{2}$), complementing a previous result that $O(n\mathrm{/}{\epsilon }^2)$ 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 $\mathrm{\Omega }(n\mathrm{/}\log n)$. We discuss implications for testing graph states and computing the generalised geometric measure of entanglement.