The Killing Form and Petz Uniqueness of Gaussian Bures Geometry
Abstract
For centered bosonic Gaussian states, the covariance pullback of the Bures cometric differs from the lift-normalized classical covariance Fisher--Rao cometric by a state-independent bilinear form. Under the canonical identification of symmetric covariance covectors with sp(2N,ℝ), this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with m pure modes has an m2-dimensional cometric kernel isomorphic to u(m), while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction (ν−ℏ/2)/(ν+ℏ/2) of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.
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BibTeX
@misc{kerskens2026killing,
title = {{The Killing Form and Petz Uniqueness of Gaussian Bures Geometry}},
author = {Christian Kerskens},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.16376},
eprint = {2607.16376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16376},
}
RIS
TY - UNPB
TI - The Killing Form and Petz Uniqueness of Gaussian Bures Geometry
AU - Christian Kerskens
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2607.16376
UR - https://arxiv.org/abs/2607.16376
AB - For centered bosonic Gaussian states, the covariance pullback of the Bures cometric differs from the lift-normalized classical covariance Fisher--Rao cometric by a state-independent bilinear form. Under the canonical identification of symmetric covariance covectors with sp(2N,ℝ), this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with m pure modes has an m^2-dimensional cometric kernel isomorphic to u(m), while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction (ν−ℏ/2)/(ν+ℏ/2) of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.
ER -