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 \(η\), the Bures--Wasserstein determinant joins them only at \((x,η)=(\sqrt\varphi,\sqrt\varphi)\). Equal determinant density does not mean tensor equality: relative to Fisher--Rao, the Bures cometric at the junction is \[
Q_{\rm gold}=\varphi^{-2}Π_0+\varphiΠ_2,
\qquad \det Q_{\rm gold}=1, \] 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.
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 (η), the Bures--Wasserstein determinant joins them only at ((x,η)=(\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}Π_0+\varphiΠ_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.
We compare four local geometries of centered one-mode bosonic Gaussiancovariances while keeping their operational meanings distinct. In a commontrace-orthonormal normalization, the Bures and covariance Fisher--Raocometric determinants cross intrinsically at x2=φ. Afterintroducing a constant action-matching parameter η, theBures--Wasserstein determinant joins them only at(x,η)=(φ,φ). Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction isQgold=φ−2Π0+φΠ2,detQgold=1,so its inverse has the exact scalar-to-weight-two quadratic penalty ratioφ3. The normalized Bogoliubov--Kubo--Mori metric supplies anindependent relative-entropy diagnostic. It agrees radially with Bures buthas shape cometric eigenvalue4ℏ2x/log[(x+1)/(x−1)]; it therefore does not form a fourthdeterminant junction and becomes tangentially stiff at the pure-stateboundary. Operationally, weight-two motion is generated by a system-onlyGaussian unitary, whereas radial motion requires a Gaussian channel orancillary dilation. The known Uhlmann--Ruan purification quotient realizesthe local Bures state metric, but an explicitsp(4,R) test shows that an unweighted right-invariantFrobenius gate norm does not reproduce the golden penalty. Conditionalbranch selection and finite-rate Bures--Wasserstein cost separation arepresented as secondary statements, not as dynamical phase transitions.
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.
We construct the covariance-sector Bures geometry of centered bosonic Gaussian states from the Gaussian symmetric-logarithmic-derivative equation and determine its quantum-admissible domain. At the Williamson uncertainty floor, the boundary geometry is anisotropic: spectrum-changing radial coefficients diverge, whereas squeezing and rotation directions along the pure Gaussian orbit remain finite. Dually, the covariance cometric acquires an exact m2-dimensional kernel when m modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential Φ=−21∑klog(νk2−41), which generates fixed-frame covariance dilation and equals βF on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is O(ℏ2/ν2), recovering covariance Fisher--Rao geometry. This intrinsic classical regime differs from the pure-state boundary, where classical statistics require a specified measurement channel. As a secondary comparison, the normalized Bures, Fisher--Rao, and action-matched Bures--Wasserstein determinant densities possess an exact one-mode junction at (x,η)=(φ,φ),x=2ν/ℏ. This junction is kinematic, and its branch interpretation is conditional. At a prescribed finite nonzero compression rate, the Bures action diverges while the Bures--Wasserstein cost remains finite; a bounded Bures budget instead enforces radial deceleration.
We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator G compatible with the Hilbert bimodule structure of a noncommutative differential calculus $\partial\colon\M\to\Hcal$ can be absorbed into the calculus itself, ∂G:=G1/2∂. The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by ∂G, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between G and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.
We construct the covariance-sector Bures geometry of centered bosonic Gaussianstates directly from the Gaussian symmetric-logarithmic-derivative equationand determine its global quantum domain. The Robertson--Schr\"odingercondition bounds this domain below by the Williamson uncertainty floor. Itsboundary geometry is anisotropic: spectrum-changing radial coefficientsdiverge, whereas squeezing and rotation coefficients remain finite along thepure Gaussian orbit. Dually, the covariance cometric develops an exactm2-dimensional kernel when m Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potentialΦ=−21∑klog(νk2−41), which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfiesΦ=βF on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed asO(ℏ2/ν2), yielding an intrinsic high-noise recovery of covarianceFisher--Rao geometry. This limit is distinct from the Williamson floor, whichis a quantum pure-state boundary rather than a classical limit. A classicalmodel at the floor instead requires an explicitly supplied measurement channelor commutative restriction; the boundary geometry constrains the structure onwhich that operational map acts but does not select the map itself. As asecondary comparison, we place the Gaussian Bures geometry alongside theFisher--Rao and Bures--Wasserstein covariance geometries. After introducing aconstant action-matching parameter η, their normalized determinantdensities possess an exact one-mode junction at(x,η)=(φ,φ). The equality is kinematic: it isneither an additional boundary nor a dynamical transition, and branch selectionrequires an explicitly stated minimum-log-determinant hypothesis. For the sameprescribed finite-rate radial compression toward the floor, the Bures kineticdensity and action diverge while the Bures--Wasserstein cost remains finite;under a bounded Bures kinetic budget, the intrinsic consequence is radialdeceleration rather than a forced change of geometry.
Finite-dimensional closed-system gate synthesis is a geometric optimal-control problem on a compact Lie group, but global solutions require careful treatment of logarithm branches, determinant-one constraints, drift, anisotropic penalties, and amplitude limits. This paper assembles and extends a self-contained family of exactly solved benchmarks under explicit hypotheses. For a positive right-invariant quadratic metric, smooth stationary curves obey the Euler–Arnold equation and possess Lax invariants. Under fully actuated isotropic control on SU(n), the fixed-time minimum action is the squared Hilbert–Schmidt distance divided by twice the gate time and is generated by a minimum-norm skew-Hermitian logarithm. A finite eigenphase-unwrapping rule computes that logarithm, while the affine spectral-width cut wall and a branch-gap criterion distinguish stable selection from set-valued behavior. Drift is removed isometrically whenever the metric is invariant under the drift adjoint action. Weighted action and box-constrained time on maximal tori and arbitrary closed commuting subtori reduce to explicit period-lattice problems; the latter gives distinct strict determinant-one and projective period lattices for an ideal fSim sector. Numerical validation uses 1506 Haar-random targets through dimension 128, a complete cost-bounded dynamic-program cross-check for every target, 300 direct Cartesian branch enumerations, 120 random-conjugation tests with analytically prescribed spectra, 240 independently cross-checked random weighted-torus instances, a separate SU(256) stress target, large-cluster selector regressions, repeated timing trials, GRAPE–L-BFGS, a first-order sequential Krotov-type update with exact discrete propagation and Fréchet derivatives, randomized and drift-adapted CRAB–Powell bases, and cut-locus tests that exercise the implemented Schur/logarithm solver. When sparse actuation, decoherence, leakage, or uncertainty invalidates the ideal hypotheses, the exact results are positioned as ideal-model reference values, branch-aware seeds, and regression tests rather than as certificates for the enlarged objective.