Finite-copy risk for polynomial energies of thermal Gaussian states
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Abstract
We study how many independent copies of a centered gauge-invariant thermal Gaussian state are needed to estimate a homogeneous normally ordered polynomial energy. General quantum symmetry averaging provides an unbiased observable; here we determine its exact finite-copy risk, including all contraction orders and the dependence on the polynomial's coefficient tensor. The construction is valid even when the number of copies is smaller than the degree, and the estimator is a spectral measurement of the averaged observable. A lower spectral bound on the occupation covariance gives a sharp dimension-independent worst-case variance, attained by a single-mode monomial, and the exact worst-polynomial minimax risk among globally unbiased estimators. Allowing bias, matching universal bounds identify the joint degree, copy-count and occupation scale for consistent estimation; in the opposite regime the minimax relative risk tends to one. Worked examples show how multimode coefficient structure reduces variance and how the uniform bounds translate into sufficient copy budgets. These results provide finite-resource benchmarks for high-order thermal-state moment estimation.
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Research details
- Subjects
- Mathematics · Quantum Information Theory · Statistics and Data Science · Statistical Theory
- Keywords
- thermal Gaussian states, quantum estimation, polynomial moments, minimax risk, symmetry averaging, occupation covariance, growing-degree estimation
- Manuscript date
- 2026-10-30
- Reader reuse
- none
Scope
Centered gauge-invariant thermal Gaussian states with a positive spectral occupation floor. Exact all-copy reduced-tensor variance, sharp worst-polynomial unbiased minimax risk, and uniform degree/copy/occupation consistency and risk-collapse laws under collective quantum measurements.
Disclosures
OpenAI Codex was used for mathematical, computational and editorial assistance in preparing this work.
Reproduction
The nine-page manuscript contains complete analytic proofs, worked photon-count and two-mode product examples, and sufficient copy budgets. The source package includes editable LaTeX, a Python standard-library verifier with 77 exact moment checks and two copy-budget calculations, portable Tectonic build instructions, citation metadata and a member checksum manifest.
References and related resources
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- R. Nair and M. Tsang, Quantum optimality of photon counting for temperature measurement of thermal astronomical sources, Astrophysical Journal 808 (2015), 125. · references
- S. Schubr, Exact minimax estimation of homogeneous complex Gaussian energies, author-published edition 2, Luminary Oddities (2026). · references
- M. Tsang, Operational meanings of a generalized conditional expectation in quantum metrology, Quantum 7 (2023), 1162. · references
- M. Tsang, F. Albarelli and A. Datta, Quantum semiparametric estimation, Physical Review X 10 (2020), 031023. · references