Probability-sensitive product probes and diagonal correction

We consider relative trace estimation of an arbitrary real positive-semidefinite matrix of order $N=n^q$ when an oracle returns the full matrix-vector response only on real product inputs. Two new upper bounds are established. A fractional-moment analysis of product-spherical empirical averaging, based on a convex-order domination of the normalized quadratic form by a product of scaled Beta variables, improves the fixed-accuracy exponential rate from $\log(2n/(n+1))$ to $D_n=\log n+1-H_n$, where $H_n$ is the $n$-th harmonic number. The rate $D_n$ is sharp for that empirical-average estimator, but it is not a minimax lower bound for the full oracle. Separately, a two-stage diagonal-control estimator has variance at most $2N\|M\|_F^2/(ms)$ after $m$ training queries and $s$ coordinate corrections, giving $O(\sqrt N/\varepsilon)$ queries with finite-bit, nonadaptive real inputs. Both results hold for every PSD input. The remaining gap between the unrestricted lower and upper bounds is stated explicitly.