Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality
Abstract
We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on n observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect E, the corresponding posterior predictive state is μE=(I+nρE)/(n+d), where ρE is the normalized one-particle marginal of E. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is hd((n+1)/(n+d)), where hd(r)=−rlog r−(1−r)log((1−r)/(d−1)). The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum ((n+1)2+d−1)/(n+d)2.
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BibTeX
@misc{hayashi2026haarbayesian,
title = {{Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality}},
author = {Masahito Hayashi and Ayanava Dasgupta and Naqueeb Ahmad Warsi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10072},
eprint = {2609.10072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10072},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality
AU - Masahito Hayashi
AU - Ayanava Dasgupta
AU - Naqueeb Ahmad Warsi
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10072
UR - https://arxiv.org/abs/2609.10072
AB - We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on n observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect E, the corresponding posterior predictive state is μ_E=(I+nρ_E)/(n+d), where ρ_E is the normalized one-particle marginal of E. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is h_d((n+1)/(n+d)), where h_d(r)=−rlog r−(1−r)log((1−r)/(d−1)). The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum ((n+1)^2+d−1)/(n+d)^2.
ER -