Trace characterizations and quantitative centrality via alternative mean inequalities

We study trace characterizations arising from comparisons between arithmetic, harmonic, and alternative means of positive definite matrices. For arithmetic mean comparisons, we prove that every alternative mean lying strictly below the arithmetic mean yields a characterization of tracial positive linear functionals. We further show that the Bures–Wasserstein midpoint is the unique alternative mean dominated by the arithmetic mean in the Loewner order, explaining why it is the only exceptional case. For harmonic mean comparisons, we introduce the harmonic obstruction threshold, a new quantity that measures the failure of trace characterization and leads to quantitative centrality estimates for positive linear functionals. As an application, we prove that, for the spectral geometric mean, every nonzero density matrix satisfying the harmonic mean comparison has condition number at most 3 + 2 2 3+2√2 . Thus, while arithmetic mean comparisons provide complete trace characterizations, harmonic mean comparisons yield a quantitative centrality principle.