Uniqueness of the averaging measure in the three-matrix Sutter--Berta--Tomamichel trace inequality

Abstract

Sutter, Berta, and Tomamichel asked whether the averaging density \(\beta_0\) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension \(d\geq2\), if a Borel probability measure \(\mu\) satisfies the averaged inequality for all \(d\times d\) Hermitian triples, then \(\mathrm d\mu(t)=\beta_0(t)\,\mathrm d t\). The full rigidity is already detected in dimension two. No moment or regularity assumption is needed. The proof combines an exact, dimension-free Fourier representation of the averaged term with a second-order perturbation of the commuting configuration \(A=C=-B\), which determines the characteristic function of \(\mu\) at every frequency.