Exact Posterior Prediction from Product Haar Measurements and a Randomized-Mesh Maximum-Likelihood Bridge

arXiv:2610.00117v1 Announce Type: new Abstract: We study prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after independently measuring the observed copies with the one-copy Haar POVM. For the resulting fixed separable observation, the Bayes predictive state under quantum relative-entropy loss is the full-rank posterior mean. We evaluate it exactly through permanents of minors of the outcome Gram matrix, prove positivity and normalization of the permanent representation, and establish minimaxity among decision rules based on these fixed outcomes. Using Hilbert--Schmidt projection of the posterior mean and an exactly analyzed linear-inversion competitor, we prove a finite-sample posterior-purity bound and an explicit relative-entropy guarantee with sample count $O((d^2/\epsilon)\log(d/\epsilon))$. Using the exact collective benchmarks proved in a companion paper, we also obtain entropy and posterior-purity comparisons. To connect the fine product data to a regular ray-valued experiment without assuming deterministic finite-sample uniqueness, we draw one random orientation for a finite projective mesh, retain it for the entire sample, and apply a selected finite-alphabet MLE in the resulting reference experiment. A nested recovering mesh uniformly recovers the fine Fisher information. Consequently every sufficiently fine fixed mesh gives a regular finite-alphabet model. Its selected MLE has Haar-averaged inverse-Fisher coefficient $\overline a_k$, and refining the mesh after the fixed-mesh large-sample limit makes $\overline a_k$ approach $d-1$. This yields epsilon-optimal leading relative-entropy and posterior-overlap bounds for the fine posterior and the sharper fixed-dimensional asymptotic scale $(d/\epsilon)\log(d/\epsilon)$. The latter is not asserted as a uniform finite-sample guarantee. No deterministic finite-sample MLE uniqueness or triangular mesh schedule is used.