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arXiv:2607.16376

The Quantum Correction to Gaussian Information Geometry is the Killing Form of the Symplectic Algebra

Christian Kerskens

quant-phmath-phSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
v1 · Jul 17, 2026v2 · Sep 14, 2026
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Abstract

On the admissible cone $\mathcal C_Ω=\{Σ\in\mathrm{Sym}^+(2n):Σ+iΩ\ge0\}$ of Gaussian covariance matrices, the classical Bures--Wasserstein transport metric, the Fisher--Rao information metric, and the quantum Bures metric are individually well understood, but their mutual relationship is obscured by the operator equations defining them. Working with the contravariant (dual) metrics, we show that the exact difference between the dual quantum Bures metric and the dual Fisher--Rao metric is independent of the covariance matrix: it is identically the trace form, proportional to the Killing form, of the symplectic algebra $\mathfrak{sp}(2n,\mathbb R)$, pulled back through the isomorphism $X\mapstoΩX$. The signature of this form on the Cartan decomposition reproduces the anisotropic stiffening of the quantum metric at the pure-state boundary: it is negative on the compact subalgebra $\mathfrak u(n)$, which carries the divergence, and positive on the noncompact complement. Because the correction is quadratic in the momenta, a no-go lemma shows it cannot arise from minimal coupling to a principal connection. We realize it instead through a Schur complement of a pseudo-Riemannian metric on a phase bundle, whose base reduction is the quantum Bures metric and whose horizontal metric is the classical-limit Fisher--Rao metric. All identities are verified numerically to working precision.

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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 exact\(m^2\)-dimensional kernel when \(m\) Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potential\(\Phi=-\tfrac12\sum_k\log(\nu_k^2-\tfrac14)\), which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfies\(\Phi=\beta F\) on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed as\(O(\hbar^2/\nu^2)\), 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 \(\eta\), their normalized determinantdensities possess an exact one-mode junction at\((x,\eta)=(\sqrt{\varphi},\sqrt{\varphi})\). 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.

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The Global Geometry of the Gaussian Bures Manifold: Admissible Domain, Spectral Boundary, and Asymptotic Classicality

Christian Kerskens

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 \(m^2\)-dimensional kernel when \(m\) modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential \[ Φ=-\tfrac12\sum_k\log(ν_k^2-\tfrac14), \] which generates fixed-frame covariance dilation and equals \(βF\) on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is \(O(\hbar^2/ν^2)\), 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 \[ (x,η)=(\sqrt{\varphi},\sqrt{\varphi}),\qquad x=2ν/\hbar. \] 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.

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

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Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra \(\mathfrak{sp}(4,\mathbb R)\). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\hbar^2(u-1)]^{-1}$ and $G_2=[\hbar^2(3u-1)]^{-1}$, where $u=(2ν/\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(β/α)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $β/α=1/u$. At the Bures-Fisher determinant crossing \(u=\varphi\), pointwise matching is possible only by inserting $β/α=\varphi^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

quant-pharXiv:2609.17574•Aug 3, 2026•0 citations

Scale freedom and thermodynamic covariance regulation in constitutive Bures geometry

Christian Kerskens

Building on the recently established decomposition of the Gaussian Bures cometric into a calibrated Fisher term and a state-independent Killing-form correction, we investigate the material realization and thermodynamic stabilization of constitutive Bures geometry. The associated Schur realization permits an independently specified action scale, whose material calibration remains a separate physical requirement. First, we show that passive static elimination of a positive symmetric auxiliary block cannot produce the required mixed-sign correction to the fixed Fisher baseline.A successful realization must therefore depart from at least one assumption of this construction. We then formulate identification of the Bures cometric as an explicit control-matching problem demanding an exact tensor match. For an active neural--flow candidate combining a cardiac pressure command, flow feedback and vascular stiffness modulation, a specified quadratic-probe diagnostic reveals limited local reachability and persistent response asymmetry in the tested parameter range. This motivates a search for independently effective controls, consistent with Ashby's principle of requisite variety.Separately, on the prescribed constitutive covariance domain, we construct an explicit GENERIC stabilization framework. Combining Hamiltonian Bures-Wasserstein kinetics, reservoir friction and a separate positive regulatory mobility, we prove local exponential stability of a prescribed interior covariance under explicit positivity and coupling conditions.While a matching physiological realization remains to be established, the framework provides a thermodynamically admissible architecture for regulating covariance dynamics on a prescribed constitutive Bures domain, opening a concrete route to investigate biological information processing without assuming a microscopic quantum substrate. Compact rotation is retained separately from covariance motion, distinguishing reversible circulation from driven dissipative circulation. Finally, a golden operating window is motivated by determinant equality and conditional gate-cost matching; a computational advantage remains a task-dependent hypothesis.

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Even Odd Splitting of the Gaussian Quantum Fisher Information: From Symplectic Geometry to Metrology

Kaustav Chatterjee, Tanmoy Pandit, Varinder Singh +2 more

We introduce a canonical decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into two additive pieces: an even part that captures changes in the symplectic spectrum and an odd part associated with correlation-generating dynamics. On the pure-state manifold, the even contribution vanishes identically, while the odd contribution coincides with the QFI derived from the natural metric on the Siegel upper half-space, revealing a direct geometric underpinning of pure-Gaussian metrology. This also provides a link between the graphical representation of pure Gaussian states and an explicit expression for the QFI in terms of graphical parameters. For evolutions completely generated by passive Gaussian unitaries (orthogonal symplectics), the odd QFI vanishes, while thermometric parameters contribute purely to the even sector with a simple spectral form; we also derive a state-dependent lower bound on the even QFI in terms of the purity-change rate. We extend the construction to the full QFI matrix, obtaining an additive even odd sector decomposition that clarifies when cross-parameter information vanishes. Applications to unitary sensing (beam splitter versus two-mode squeezing) and to Gaussian channels (loss and phase-insensitive amplification), including joint phase loss estimation, demonstrate how the decomposition cleanly separates resources associated with spectrum versus correlations. The framework supplies practical design rules for continuous-variable sensors and provides a geometric lens for benchmarking probes and channels in Gaussian quantum metrology.

quant-pharXiv:2601.06513•Jan 13, 2026•0 citations
On the admissible cone CΩ={Σ∈Sym+(2n):Σ+iΩ≥0}\mathcal C_Ω=\{Σ\in\mathrm{Sym}^+(2n):Σ+iΩ\ge0\}CΩ​={Σ∈Sym+(2n):Σ+iΩ≥0} of Gaussian covariance matrices, the classical Bures--Wasserstein transport metric, the Fisher--Rao information metric, and the quantum Bures metric are individually well understood, but their mutual relationship is obscured by the operator equations defining them. Working with the contravariant (dual) metrics, we show that the exact difference between the dual quantum Bures metric and the dual Fisher--Rao metric is independent of the covariance matrix: it is identically the trace form, proportional to the Killing form, of the symplectic algebra sp(2n,R)\mathfrak{sp}(2n,\mathbb R)sp(2n,R), pulled back through the isomorphism X↦ΩXX\mapstoΩXX↦ΩX. The signature of this form on the Cartan decomposition reproduces the anisotropic stiffening of the quantum metric at the pure-state boundary: it is negative on the compact subalgebra u(n)\mathfrak u(n)u(n), which carries the divergence, and positive on the noncompact complement. Because the correction is quadratic in the momenta, a no-go lemma shows it cannot arise from minimal coupling to a principal connection. We realize it instead through a Schur complement of a pseudo-Riemannian metric on a phase bundle, whose base reduction is the quantum Bures metric and whose horizontal metric is the classical-limit Fisher--Rao metric. All identities are verified numerically to working precision.
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 exactm2m^2m2-dimensional kernel when mmm Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potentialΦ=−12∑klog⁡(νk2−14)\Phi=-\tfrac12\sum_k\log(\nu_k^2-\tfrac14)Φ=−21​∑k​log(νk2​−41​), which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfiesΦ=βF\Phi=\beta FΦ=βF on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed asO(ℏ2/ν2)O(\hbar^2/\nu^2)O(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\etaη, their normalized determinantdensities possess an exact one-mode junction at(x,η)=(φ,φ)(x,\eta)=(\sqrt{\varphi},\sqrt{\varphi})(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.
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 m2m^2m2-dimensional kernel when mmm modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential Φ=−12∑klog⁡(νk2−14), Φ=-\tfrac12\sum_k\log(ν_k^2-\tfrac14), Φ=−21​∑k​log(νk2​−41​), which generates fixed-frame covariance dilation and equals βFβFβF on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is O(ℏ2/ν2)O(\hbar^2/ν^2)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. (x,η)=(\sqrt{\varphi},\sqrt{\varphi}),\qquad x=2ν/\hbar. (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 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=φx^2=\varphix2=φ. Afterintroducing a constant action-matching parameter η\etaη, theBures--Wasserstein determinant joins them only at(x,η)=(φ,φ)(x,\eta)=(\sqrt\varphi,\sqrt\varphi)(x,η)=(φ​,φ​). Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction isQgold=φ−2Π0+φΠ2,det⁡Qgold=1, Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1,Qgold​=φ−2Π0​+φΠ2​,detQgold​=1,so its inverse has the exact scalar-to-weight-two quadratic penalty ratioφ3\varphi^3φ3. 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)]4\hbar^2x/\log[(x+1)/(x-1)]4ℏ2x/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)\mathfrak{sp}(4,\mathbb R)sp(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.
Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra sp(4,R)\mathfrak{sp}(4,\mathbb R)sp(4,R). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are G0=[ℏ2(u−1)]−1G_0=[\hbar^2(u-1)]^{-1}G0​=[ℏ2(u−1)]−1 and G2=[ℏ2(3u−1)]−1G_2=[\hbar^2(3u-1)]^{-1}G2​=[ℏ2(3u−1)]−1, where u=(2ν/ℏ)2u=(2ν/\hbar)^2u=(2ν/ℏ)2. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every uuu; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives G0/G2=1+2(β/α)u/(u−1)G_0/G_2=1+2(β/α)u/(u-1)G0​/G2​=1+2(β/α)u/(u−1); matching Bures throughout the isotropic family would require the state-dependent relation β/α=1/uβ/α=1/uβ/α=1/u. At the Bures-Fisher determinant crossing u=φu=\varphiu=φ, pointwise matching is possible only by inserting β/α=φ−1β/α=\varphi^{-1}β/α=φ−1. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.
Building on the recently established decomposition of the Gaussian Bures cometric into a calibrated Fisher term and a state-independent Killing-form correction, we investigate the material realization and thermodynamic stabilization of constitutive Bures geometry. The associated Schur realization permits an independently specified action scale, whose material calibration remains a separate physical requirement. First, we show that passive static elimination of a positive symmetric auxiliary block cannot produce the required mixed-sign correction to the fixed Fisher baseline.A successful realization must therefore depart from at least one assumption of this construction. We then formulate identification of the Bures cometric as an explicit control-matching problem demanding an exact tensor match. For an active neural--flow candidate combining a cardiac pressure command, flow feedback and vascular stiffness modulation, a specified quadratic-probe diagnostic reveals limited local reachability and persistent response asymmetry in the tested parameter range. This motivates a search for independently effective controls, consistent with Ashby's principle of requisite variety.Separately, on the prescribed constitutive covariance domain, we construct an explicit GENERIC stabilization framework. Combining Hamiltonian Bures-Wasserstein kinetics, reservoir friction and a separate positive regulatory mobility, we prove local exponential stability of a prescribed interior covariance under explicit positivity and coupling conditions.While a matching physiological realization remains to be established, the framework provides a thermodynamically admissible architecture for regulating covariance dynamics on a prescribed constitutive Bures domain, opening a concrete route to investigate biological information processing without assuming a microscopic quantum substrate. Compact rotation is retained separately from covariance motion, distinguishing reversible circulation from driven dissipative circulation. Finally, a golden operating window is motivated by determinant equality and conditional gate-cost matching; a computational advantage remains a task-dependent hypothesis.
We introduce a canonical decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into two additive pieces: an even part that captures changes in the symplectic spectrum and an odd part associated with correlation-generating dynamics. On the pure-state manifold, the even contribution vanishes identically, while the odd contribution coincides with the QFI derived from the natural metric on the Siegel upper half-space, revealing a direct geometric underpinning of pure-Gaussian metrology. This also provides a link between the graphical representation of pure Gaussian states and an explicit expression for the QFI in terms of graphical parameters. For evolutions completely generated by passive Gaussian unitaries (orthogonal symplectics), the odd QFI vanishes, while thermometric parameters contribute purely to the even sector with a simple spectral form; we also derive a state-dependent lower bound on the even QFI in terms of the purity-change rate. We extend the construction to the full QFI matrix, obtaining an additive even odd sector decomposition that clarifies when cross-parameter information vanishes. Applications to unitary sensing (beam splitter versus two-mode squeezing) and to Gaussian channels (loss and phase-insensitive amplification), including joint phase loss estimation, demonstrate how the decomposition cleanly separates resources associated with spectrum versus correlations. The framework supplies practical design rules for continuous-variable sensors and provides a geometric lens for benchmarking probes and channels in Gaussian quantum metrology.