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arXiv · 2406.16827

Testing multipartite productness is easier than testing bipartite productness

Benjamin D. M. Jones·Ashley Montanaro
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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.

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Benjamin D. M. Jones, Ashley Montanaro. 2024-06-24. Testing multipartite productness is easier than testing bipartite productness. https://arxiv.org/abs/2406.16827

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