Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka–Hayashi Bound
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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