Schurunicodex2013Weyl Reduction of Finite-Copy Precision Bounds for Multiparameter Quantum Estimation
Abstract
In multiparameter quantum estimation, incompatible measurements can impose trade-offs among estimation errors. Understanding the precision attainable with a finite number of jointly measured copies is therefore important, but direct optimization becomes costly as the tensor product space grows exponentially. For independent and identically distributed (i.i.d.) copies, the quantum statistical model is invariant under permutations of the copies. We exploit this symmetry through Schur--Weyl duality to remove permutation multiplicity spaces while using their dimensions as weights. This gives a reduced statistical model equivalent to the original under arbitrary collective measurements. On this reduced model, we derive an exact conic formulation of the tight Cramér--Rao bound and a semidefinite program for its Nagaoka--Hayashi (NH) relaxation. For fixed single-copy dimension, the reduced operator representation grows polynomially rather than exponentially with the number of copies. To approximate the tight bound, we use an exact quadratic identity to correct finite-ray dual solutions and obtain rigorous upper and lower bounds. For fixed copy number and parameter dimension, sufficiently fine meshes admit feasible primal points, and dual solutions with bounded objective gaps have uniformly bounded compact-search radii. These results yield quadratic convergence in the projective covering radius. Numerical examples illustrate the quadratic scaling of the mesh correction term and the computational reach of the reduced NH formulation.
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BibTeX
@misc{zhang2026schurunicodex2013weyl,
title = {{Schur$\unicodex2013$Weyl Reduction of Finite-Copy Precision Bounds for Multiparameter Quantum Estimation}},
author = {Jianchao Zhang and Masahito Hayashi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38299},
eprint = {2610.38299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38299},
}
RIS
TY - UNPB
TI - Schurunicodex2013Weyl Reduction of Finite-Copy Precision Bounds for Multiparameter Quantum Estimation
AU - Jianchao Zhang
AU - Masahito Hayashi
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38299
UR - https://arxiv.org/abs/2610.38299
AB - In multiparameter quantum estimation, incompatible measurements can impose trade-offs among estimation errors. Understanding the precision attainable with a finite number of jointly measured copies is therefore important, but direct optimization becomes costly as the tensor product space grows exponentially. For independent and identically distributed (i.i.d.) copies, the quantum statistical model is invariant under permutations of the copies. We exploit this symmetry through Schur--Weyl duality to remove permutation multiplicity spaces while using their dimensions as weights. This gives a reduced statistical model equivalent to the original under arbitrary collective measurements. On this reduced model, we derive an exact conic formulation of the tight Cramér--Rao bound and a semidefinite program for its Nagaoka--Hayashi (NH) relaxation. For fixed single-copy dimension, the reduced operator representation grows polynomially rather than exponentially with the number of copies. To approximate the tight bound, we use an exact quadratic identity to correct finite-ray dual solutions and obtain rigorous upper and lower bounds. For fixed copy number and parameter dimension, sufficiently fine meshes admit feasible primal points, and dual solutions with bounded objective gaps have uniformly bounded compact-search radii. These results yield quadratic convergence in the projective covering radius. Numerical examples illustrate the quadratic scaling of the mesh correction term and the computational reach of the reduced NH formulation.
ER -