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

2026/07/17 by Christian Kerskens
#quant-ph #math-ph #math.MP

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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[Σ], where L_Σ is the Lyapunov operator, and explicit Hamiltonian 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.

Citations

• Stochastic thermodynamics, fluctuation theorems and molecular machines
• Gaussian quantum information
• A well-conditioned estimator for large-dimensional covariance matrices

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