Classical Sufficiency in Quantum Statistical Experiments
Abstract
A quantum measurement transforms a quantum statistical experiment into a classical one, but different measurements generally yield experiments with different statistical information. We introduce quantum-to-classical sufficiency: relative to a prescribed measurement class, a measurement is sufficient if its induced classical experiment Blackwell-dominates those from every other measurement in the class. This separates two components of quantum inference: reducing the measurement class through symmetry or decision-theoretic arguments, and identifying a canonical measurement in the reduced class by an experiment-level sufficiency condition. We establish a factorization criterion where the parameter enters the group-averaged experiment only through weights of an orthogonal block decomposition. The block-label measurement is then sufficient relative to all invariant measurements. For tensor-product models under unitary conjugation, this identifies weak Schur sampling as the sufficient measurement for spectral inference. Combined with risk-preserving symmetrization, it reduces invariant Bayes and minimax problems over all measurements exactly to classical decision problems based on Young diagrams. We derive sharp first-order asymptotic Bayes and minimax risks for smooth spectral functionals. We establish an analogous reduction for a noncompact group action in a continuous-variable setting. We consider displaced thermal states and functionals of the thermal parameter with an unknown displacement. Within displacement-invariant measurements, total residual photon number measurement emerges as the sufficient measurement. Combined with Hunt-Stein reduction, this yields exact finite-sample reductions of Bayes-minimax and minimax problems over all measurements to classical decision problems based on the total residual photon count, which follows a negative binomial distribution.
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BibTeX
@misc{lahiry2026classical,
title = {{Classical Sufficiency in Quantum Statistical Experiments}},
author = {Samriddha Lahiry},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29348},
eprint = {2609.29348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29348},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classical Sufficiency in Quantum Statistical Experiments
AU - Samriddha Lahiry
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29348
UR - https://arxiv.org/abs/2609.29348
AB - A quantum measurement transforms a quantum statistical experiment into a classical one, but different measurements generally yield experiments with different statistical information. We introduce quantum-to-classical sufficiency: relative to a prescribed measurement class, a measurement is sufficient if its induced classical experiment Blackwell-dominates those from every other measurement in the class. This separates two components of quantum inference: reducing the measurement class through symmetry or decision-theoretic arguments, and identifying a canonical measurement in the reduced class by an experiment-level sufficiency condition. We establish a factorization criterion where the parameter enters the group-averaged experiment only through weights of an orthogonal block decomposition. The block-label measurement is then sufficient relative to all invariant measurements. For tensor-product models under unitary conjugation, this identifies weak Schur sampling as the sufficient measurement for spectral inference. Combined with risk-preserving symmetrization, it reduces invariant Bayes and minimax problems over all measurements exactly to classical decision problems based on Young diagrams. We derive sharp first-order asymptotic Bayes and minimax risks for smooth spectral functionals. We establish an analogous reduction for a noncompact group action in a continuous-variable setting. We consider displaced thermal states and functionals of the thermal parameter with an unknown displacement. Within displacement-invariant measurements, total residual photon number measurement emerges as the sufficient measurement. Combined with Hunt-Stein reduction, this yields exact finite-sample reductions of Bayes-minimax and minimax problems over all measurements to classical decision problems based on the total residual photon count, which follows a negative binomial distribution.
ER -