Characterize trace super-uniform random matrices

Determine a complete characterization of all S_d^+-valued random matrices U that are trace super-uniform, meaning they satisfy P(U ⪯ Y) ≤ tr(Y) for every Y ∈ S_d^+, and, when applicable, identify subclasses such as Y-trace uniform distributions for which equality holds on a specified family Y ⊆ S_d^+.

Background

The paper introduces trace super-uniform random matrices U to obtain randomized matrix Markov inequalities that strictly tighten classical bounds. A matrix U is called trace super-uniform if P(U ⪯ Y) ≤ tr(Y) for all Y in S_d^+. Examples include UI when U ~ Unif(0,1) and some shifted versions UI + Y with Y ⪰ 0, but a general description of all such matrices is not known.

A full characterization would clarify the allowable randomizers that preserve validity while maximizing tightness across different matrix tail problems, and would inform when equality conditions hold (Y-trace uniformity).