When the noncommutative AM-GM inequality holds

In this note, we prove that the noncommutative AM-GM inequality holds if $n\ge 2\lceil m/2 \rceil^2$. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.