Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka–Hayashi Bound
Source: hexagonmath.org/abs/2610.00522v1
Bayesian weak commutativity has been suggested as the most likely condition for reaching the Nagaoka–Hayashi (NH) bound in multiparameter estimation. We show that it is not sufficient. For a full-rank four-level ensemble obeying it exactly, the effects required for NH equality cannot sum to the identity. We find the optimal risk exactly: it lies strictly above the bound and needs seven outcomes, while projective readout on the four levels adds over seven times the NH gap. A robust normalization witness bounds the gap from below across a family including truncated oscillator ladders.
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