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arXiv:2607.16380

Hamiltonian Lift of Bures--Wasserstein Covariance Dynamics with a Spectral Floor

Christian Kerskens

quant-phmath-phSubmitted Jul 17, 2026• Announced Jul 20, 2026
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Abstract

Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Sym$^+_n$ equipped with the Bures--Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum $Π=\tfrac12 L_Σ[\dotΣ]$, where $L_Σ$ is the Lyapunov operator, and explicit Hamiltonian $\mathcal{H}(Σ,Π) = 2{\rm tr}(ΠΣΠ)+V(Σ)$. Adding Rayleigh dissipation recovers the Bures--Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as $(s-ν)^{-2}$ as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures--Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.

DOI
10.1016/j.physleta.2026.132215
Comments
Physics Letter A in revision

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Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Symn+^+_nn+​ equipped with the Bures--Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum Π=12LΣ[Σ˙]Π=\tfrac12 L_Σ[\dotΣ]Π=21​LΣ​[Σ˙], where LΣL_ΣLΣ​ is the Lyapunov operator, and explicit Hamiltonian H(Σ,Π)=2tr(ΠΣΠ)+V(Σ)\mathcal{H}(Σ,Π) = 2{\rm tr}(ΠΣΠ)+V(Σ)H(Σ,Π)=2tr(ΠΣΠ)+V(Σ). Adding Rayleigh dissipation recovers the Bures--Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as (s−ν)−2(s-ν)^{-2}(s−ν)−2 as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures--Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.
Bures--Wasserstein covariance dynamics admits a conservative Hamiltonian parent whose Eisenhart--Duval lift defines a Bargmann geometry. After passing to the conformal frame associated with the parabolic z=2\mathfrak z=2z=2 scaling of the free covariance flow, we study a Schr\"odinger-type temporal deformation of this lifted geometry. The null charge of the Bargmann lift acts as an informational mass. For nonzero temporal deformation and nonzero informational mass, accessible massive null geodesics cannot reach the boundary of absolute certainty, where the covariance scale vanishes, but instead possess a finite inner turning point. Reinstating a logarithmic spectral-floor potential shows that this kinetic mass gap and the substrate capacity floor combine into a single effective radial potential, with the operative cutoff set by the larger of the two radii. Transverse squeezing momentum supplies an outer confinement and yields a loading bound: a confined orbit exists only while the conserved energy dominates the transverse momentum, and when that bound fails the radial well disappears through the merger of its turning points. We interpret this result, conditionally, as a geometric model of finite stable processing intervals in information-processing substrates, while restricting the formal claims to centred covariance dynamics in a local neighbourhood of the isotropic ray of the positive-definite cone.
The Bures--Wasserstein (BW) covariance programme developed in a series of recent preprints presents its dynamics as capacity-constrained gradient flows on the cone of positive-definite matrices. We show that the same dynamics admit a variational formulation with explicit symplectic structure. We construct a conservative action on the BW manifold, extended by capacity parameters $(\numin, \Qmax, \Dfixed)$ entering as couplings, and show that its Euler--Lagrange equations reproduce the BW dynamics after a standard Legendre transform. The symplectic form on the associated phase space restricts cleanly to the double-quantum (DQ) and zero-quantum (ZQ) subspaces previously identified in the algebraic sector classification of the programme. We prove that the quadratic part of the Hamiltonian contains a†a†+aaa^{\dagger}a^{\dagger} + aaa†a†+aa pairing terms in the DQ sector and bilinear mixing terms in the ZQ sector, and that these algebraic structures force SU(1,1) and SU(2) respectively as the kinematic symmetry groups. The sector classification is therefore not ad hoc but a consequence of the variational structure. The resulting action remains symplectically well posed across admissible capacity couplings and provides the conservative foundation for later dissipative and boundary-theoretic extensions.
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 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.
The Bures--Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three--sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton--DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ=Φ2Ψ=Φ^2Ψ=Φ2 satisfies the linear forced heat equation \begin{equation*} D_tΨ=Ψ_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures--Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature 6/(1−4t)6/(1-4t)6/(1−4t) and extinction time T=1/4T=1/4T=1/4. Thus the metric remains inside the monotone cone for all t<Tt<Tt<T and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume--normalized flow, for which the Bures--Helstrom metric is a fixed point. Its linearization is the shifted round--sphere Laplacian ΔS3+3Δ_{\mathbb S^3}+3ΔS3​+3, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap 555 after removal of the scaling mode.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but the geometric description of the latent world (modaldimension) still lacks a rigorous metric theory. This paper elevates the latent worldalgebraic framework to the level of information geometry, proving that the naturalmetric carried by the latent world subalgebra Alat is precisely the Bures-Wassersteinmetric, and that the measurement projection π : A → Aobs corresponds to a gradient flow from the Wasserstein-2 space to the Fisher-Rao space. The core discoveryis that the critical condition for the information potential difference ∆µ ≥ ln2 isequivalent to the curvature singularity of the Bures distance, and the projection dynamics is exactly the blow-up solution of the Wasserstein gradient flow at the divergence of the Ricci curvature. This paper establishes the exact relation Klat = −14FQ between the latent world curvature Klat and the quantum Fisher information FQ,proposes testable predictions of information-geometric phase transitions, and extends the framework to noncommutative geometry (spectral triples) and interfaceswith quantum gravity. This work provides a unified perspective of differential geometry and optimal transport for understanding the quantum-classical transition.