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Variational Bures--Wasserstein Dynamics under Capacity Constraint

Christian Kerskens

Published Apr 28, 2026• 0 citations
Publisher

Abstract

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 $a^{\dagger}a^{\dagger} + aa$ 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.

DOI
10.5281/zenodo.19859899

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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.
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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.
This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.
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.
Computing bosonic Gaussian ground states via variational optimization is challenging because the covariance matrices must satisfy the uncertainty principle, rendering constrained or Riemannian optimization costly, delicate, and thus difficult to scale, particularly in large and inhomogeneous systems. We introduce a symplectic optimization framework that addresses this challenge by parameterizing covariance matrices directly as positive-definite symplectic matrices using unit-triangular factorizations. This approach enforces all physical constraints exactly, yielding a globally unconstrained variational formulation of the bosonic ground-state problem. The unconstrained structure also naturally supports solution reuse across nearby Hamiltonians: warm-starting from previously optimized covariance matrices substantially reduces the number of optimization steps required for convergence in families of related configurations, as encountered in crystal lattices, molecular systems, and fluids. We demonstrate the method on weakly dipole-coupled lattices, recovering ground-state energies, covariance matrices, and spectral gaps accurately. The framework further provides a foundation for large-scale approximate treatments of weakly non-quadratic interactions and offers potential scaling advantages through tensor-network enhancements.