Testing multipartite productness is easier than testing bipartite productness

TL;DR

This paper establishes a lower bound on the number of copies needed to test if a multipartite quantum state is bipartite product, showing that testing multipartite productness is fundamentally easier than bipartite testing.

Contribution

It proves a tight lower bound of 1n / \u221aa0n copies for testing bipartite productness in multipartite states, complementing existing upper bounds.

Findings

Omega(n / log n) copies are necessary for testing bipartite productness.

Testing multipartite productness requires fewer copies than bipartite testing.

The proof uses random ensembles and trace distance analysis.

Abstract

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 $\Omega(n / \log n)$ copies are required (for fixed $\epsilon \leq \frac{1}{2}$), complementing a previous result that $O(n / \epsilon^2)$ 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 $\Omega (n / \log n)$. We discuss implications for testing graph states and computing the generalised geometric measure of entanglement.