An attainable Gill-Massar-type bound for spin-factor models
Abstract
We determine the exact local precision limits for single-copy estimation of smooth multiparameter quantum statistical models contained in spin factors, a class of matrix Jordan algebras whose state spaces generalize the qubit Bloch ball. At any parameter point where the symmetric logarithmic derivative (SLD) Fisher information is positive definite, we characterize the entire attainable classical Fisher-information region over all finite-outcome positive-operator-valued measurements. After SLD normalization, this region consists exactly of the real symmetric positive semidefinite matrices with trace at most one, independently of the ambient Hilbert-space dimension. This yields a sharp weighted covariance bound for every positive definite weight, attained by an explicit locally unbiased estimator based on randomized spectral measurements of SLD directions. The proof combines a statistics-preserving positive projection onto the spin factor with the two-eigenvalue structure of its effects, revealing the Jordan-algebraic origin of the tradeoff. The result extends the qubit information tradeoff to models with more than three parameters. Since the optimal measurement depends on the unknown parameter, we simulate an adaptive scheme for a five-parameter model on a four-dimensional Hilbert space and observe performance close to the optimal local benchmark.
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BibTeX
@misc{yamagata2026attainable,
title = {{An attainable Gill-Massar-type bound for spin-factor models}},
author = {Koichi Yamagata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23020},
eprint = {2609.23020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23020},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An attainable Gill-Massar-type bound for spin-factor models
AU - Koichi Yamagata
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23020
UR - https://arxiv.org/abs/2609.23020
AB - We determine the exact local precision limits for single-copy estimation of smooth multiparameter quantum statistical models contained in spin factors, a class of matrix Jordan algebras whose state spaces generalize the qubit Bloch ball. At any parameter point where the symmetric logarithmic derivative (SLD) Fisher information is positive definite, we characterize the entire attainable classical Fisher-information region over all finite-outcome positive-operator-valued measurements. After SLD normalization, this region consists exactly of the real symmetric positive semidefinite matrices with trace at most one, independently of the ambient Hilbert-space dimension. This yields a sharp weighted covariance bound for every positive definite weight, attained by an explicit locally unbiased estimator based on randomized spectral measurements of SLD directions. The proof combines a statistics-preserving positive projection onto the spin factor with the two-eigenvalue structure of its effects, revealing the Jordan-algebraic origin of the tradeoff. The result extends the qubit information tradeoff to models with more than three parameters. Since the optimal measurement depends on the unknown parameter, we simulate an adaptive scheme for a five-parameter model on a four-dimensional Hilbert space and observe performance close to the optimal local benchmark.
ER -