The Golden Junction of Gaussian Covariance Geometries: Sector Anisotropy, Relative-Entropy Response, and Local Penalties

Abstract:We compare four local geometries of centered one-mode bosonic Gaussian covariances while keeping their operational meanings distinct. In a common trace-orthonormal normalization, the Bures and covariance Fisher--Rao cometric determinants cross intrinsically at \(x^2=\varphi\). After introducing a constant action-matching parameter \(\eta\), the Bures--Wasserstein determinant joins them only at \((x,\eta)=(\sqrt\varphi,\sqrt\varphi)\). Equal determinant density does not mean tensor equality: relative to Fisher--Rao, the Bures cometric at the junction is \[ Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1, \] so its inverse has the exact scalar-to-weight-two quadratic penalty ratio \(\varphi^3\). The normalized Bogoliubov--Kubo--Mori metric supplies an independent relative-entropy diagnostic. It agrees radially with Bures but has shape cometric eigenvalue \(4\hbar^2x/\log[(x+1)/(x-1)]\); it therefore does not form a fourth determinant junction and becomes tangentially stiff at the pure-state boundary. Operationally, weight-two motion is generated by a system-only Gaussian unitary, whereas radial motion requires a Gaussian channel or ancillary dilation. The known Uhlmann--Ruan purification quotient realizes the local Bures state metric, but an explicit \(\mathfrak{sp}(4,\mathbb R)\) test shows that an unweighted right-invariant Frobenius gate norm does not reproduce the golden penalty. Conditional branch selection and finite-rate Bures--Wasserstein cost separation are presented as secondary statements, not as dynamical phase transitions.

arXiv:2609.17574 [quant-ph]

(or arXiv:2609.17574v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.17574 Focus to learn more arXiv-issued DOI via DataCite

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