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on Sample complexity of entanglement testing

The statement

A mixed state on two $n$ -qubit registers is separable when it is a convex combination of product states $\rho\otimes\sigma$ . Given copies of a state promised to be separable or $\varepsilon$ -far in trace distance from every separable state, determine the number of copies needed to distinguish the cases, allowing entangled measurements. The 2024 slides give a gap between a lower bound of order $2^{2n}/\varepsilon^2$ and an upper bound of order $2^{4n}/\varepsilon^2$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.