Finite-copy risk for polynomial energies of thermal Gaussian states
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Abstract
We study estimation of homogeneous normally ordered polynomial energies from independent copies of a centered gauge-invariant Gaussian state. For every positive copy count, passive mixing of the copies produces a positive unbiased observable whose relative variance has an exact formula in terms of reduced density matrices of a covariance-weighted coefficient tensor. Under a lower spectral bound on the occupation covariance, the formula gives a sharp dimension-independent worst-case variance, attained by a single-mode monomial. It is also the worst-polynomial minimax risk among globally unbiased quantum estimators. Allowing bias, we prove matching universal bounds in the degree, copy count and occupation floor. These bounds characterize consistency, and a further two-point argument proves that the minimax relative risk tends to one in the opposite regime. The construction uses quantum symmetry averaging, while its exact risk analysis combines bosonic Gaussian contractions with complex-sphere moments.
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Available editions
- Edition 1 · 2026-10-30 · public
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 eight-page manuscript contains complete analytic proofs. The source package includes editable LaTeX, a Python standard-library verifier with seven exact two-mode contraction examples and 64 scalar identities, 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