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Global Convergence and Boundary Invariance of the Riemannian Bures Gradient Flow for d = 2 and d = 3

Gervaseus Nerik

Published Oct 18, 2026• 0 citations
Publisher

Abstract

We extend the Riemannian Bures gradient-flow analysis of the competing-poles potential $V_{\mathrm{dyn}}[\rho]$ from the $d=2$ qubit Bloch ball to the 8-dimensional manifold $\mathcal{M}_\circ(\mathbb{C}^3)$ of full-rank qutrit density matrices. Using the coordinate-free contravariant inverse Bures metric $G_\rho^{-1}(H) = 2(\rho H + H\rho) - 4\operatorname{Tr}(\rho H)\rho$ and the normalized $SU(3)$ Hilbert–Schmidt purity distance $B_{\mathrm{HS}}(\rho) = \frac{1}{2}(3\operatorname{Tr}(\rho^2)-1)$, we prove the following analytical results without boundary clamping ($\epsilon = 0$): Operator Commutation: Every critical point on $\mathcal{M}_\circ(\mathbb{C}^d)$ commutes with the reference pure projector $\Pi_*$. Orientation Reduction: At fixed spectrum, the non-logarithmic outward eigenvalue pull is strictly increasing in $\sqrt{\rho_{00}}$, reducing the $SU(3)$ boundary-orientation problem to a two-parameter diagonal family. Forward Invariance: The one-sided derivative of $\lambda_{\min}(\rho)$ along the flow is strictly positive on the analytic collar $0 < \lambda_{\min} \le 6.84 \times 10^{-5}$, establishing manifold forward invariance (with sharp numerical threshold $\lambda_{\mathrm{collar}} = 0.00481018$). Transverse Uniqueness: No asymmetric critical points ($\lambda_2 \neq \lambda_3$) exist despite the transverse potential's concavity. Coaxial Attractor: The coaxial balance equation admits a unique root $p^* = 0.961593764$, $\lambda_\perp^* = 0.019203118$, $B_{\mathrm{HS}}(\rho^*) = 0.888100$. Global Convergence: By LaSalle's Invariance Principle, every trajectory initialized in $\mathcal{M}_\circ(\mathbb{C}^3)$ converges globally to $\rho^*$. Includes complete standalone Python verification suite (ntcr_d3_verify.py) and rigorous numerical cross-checks across 105 complex starts.

DOI
10.5281/zenodo.23456489

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Harmonic Maps on the Bloch Ball with the Bures Metric: Local Jets, the Global \(S^{3}\) Spectrum, and the Distortion Observable \(D, Q\)

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This repository contains the complete replication package for the research paper: "Harmonic Maps on the Bloch Ball with the Bures Metric: Local Jets, the Global $S^3$ Spectrum, and the Distortion Observable $D, Q$". We study harmonic self-maps $\Phi : \mathcal{M} \to \mathcal{M}$ on the open single-qubit Bloch ball equipped with the Bures (quantum Fisher information) metric in the sense of Eells and Sampson. Key Contributions:Complete Affine Classification: We prove that a nondegenerate affine map is harmonic if and only if it is a global isometry ($A \in O(3)$ and $t = 0$), recovering the exact boundary between unitary quantum channels ($SO(3)$) and positive but not completely positive (non-CP) maps. The Distortion Observable $D$: We introduce the Riemannian invariant $D(\Phi) := g_{\Phi(r)}(\tau, \tau)$ as a geometric distance to unitarity and evaluate it in closed form for the dephasing, depolarizing, and amplitude-damping channels. Pointwise Jacobi Jet (Block A): We compute the exact covariant closed form of the infinitesimal quadratic expansion $Q$ around the identity map at every point of the ball, proving it descends canonically to the Bures metric on the effective quotient of generators. Global $S^3$ Spectral Geometry (Block B): By establishing a global isometry between the Bloch ball and an open hemisphere of the round 3-sphere of radius $1/2$, we completely solve the global spectrum of the linearized tension (Jacobi) operator $T = \Delta^{\text{rough}}_g + 8\,\text{Id}$. Via representation theory (Schur's lemma applied to the $SU(2)_L \times SU(2)_R$ action), we determine the exact closed-form eigenvalues and multiplicities for both the gradient (Weitzenböck) and the toroidal sectors. Repository Contents (.zip file):1. Manuscript (PDF): The full text of the research article including complete derivations and theoretical proofs.2. Replication Source Code (Python/SymPy): Independently runnable scripts (relying only on `sympy`, `numpy`, and `mpmath`) that verify every closed-form identity, perform the high-precision finite-difference checks, and execute the numerical SVD diagonalization confirming the full closure of the toroidal sector at degrees $\ell = 2$ and $\ell = 3$. Keywords: harmonic maps; Bures metric; quantum Fisher information; Bloch ball; monotone metrics; Weitzenböck formula; homogeneous spaces; $S^3$ spectral geometry.

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Version 2 (25 October 2026) adds an erratum. The main results of this record check out: the inverse Bures metric, the eigenvalue equation (6), the boundary collar 0 < lambda_min < 0.0564, both tables and the attractor p* = 0.9412018 were reproduced independently. Two steps in the proof text are corrected; the 0.0564 collar is labeled as established numerically (the collar proven analytically in version 3 of the Bures paper is 0.0261); and the weights are labeled as modeling choices, because the results depend on them. Every number in the erratum is printed by the attached script (addendum_d2_checks.py), included with its saved output. Version 1 remains available unchanged. The complete d=2 analysis, with full proofs, is at https://doi.org/10.5281/zenodo.23605523 Original description (version 1): We present an exact analytical and numerical analysis of the d=2 qubit reduction and logarithmic boundary barrier dynamics within the Nerik Theory of Conscious Refinement (NTCR V4.27). Through contravariant metric derivation and 50-digit numerical verification, we establish the exact Riemannian Bures eigenvalue velocity equation, superseding earlier heuristic coaxial approximations. By formulating the two-pole structured complexity C(rho) using the smooth Hilbert-Schmidt purity distance B_HS(rho) = 2 Tr((rho - I/2)^2) = r^2 alongside the Bures distance d_B^2(rho, \Pi_*) to the pure boundary projector \Pi_* = |\psi_*><\psi_*| and the von Neumann relative entropy barrier D(rho || I/2) = ln 2 - S(\rho), we eliminate the off-axis 1/sqrt(\lambda_{min}) boundary cusp and remove all ad-hoc coherence penalties and regularization parameters (\epsilon = 0). We prove analytically that: (i) radial Bloch vectors are exact eigenvectors of the 2x2 inverse Bures metric tensor G^{-1} with eigenvalue 1 - r^2 = 4 \lambda_{min} (1 - \lambda_{min}); (ii) d\lambda_{min}/d\tau > 0 holds uniformly across all angles \theta \in [0, 2\pi) inside the boundary collar 0 < \lambda_{min} < 0.0564; and (iii) no off-axis critical points exist anywhere in the full-rank manifold \mathcal{M}_\circ, driving all interior trajectories to the unique emergent interior attractor (p^* = 0.941202, x_c^* = 0, \lambda_{min}^* = 0.058798).

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Ricci flow for the Bures--Helstrom qubit metric

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We formulate a geometric extension of open quantum dynamics in which the density matrix evolves as a Riemannian gradient flow on the Bures manifold. The evolution equation \frac{d\rho}{dt} = -i[H,\rho] - \lambda \nabla_B \Phi[\rho] drives the state toward regions of minimal information degradation, defined by the functional \Phi[\rho], the instantaneous rate of relative-entropy change under the bare Lindbladian. This deterministic flow preserves trace and positivity, and introduces a single new universal constant \lambda>0 while recovering standard Lindblad dynamics as \lambda\to0. Under a Unitary Orthogonality Condition – exactly satisfied for pure dephasing and approximately in many-body localized phases – \Phi becomes a Lyapunov functional, and its critical points (decoherence-free subspaces) are asymptotically stable attractors. Applied to disordered spin chains, the framework reproduces the known entropy‑suppression scaling \Phi \lesssim \gamma N (J/W)^2. The theory offers a falsifiable, information‑geometric principle where quantum states intrinsically minimise their own information loss. Keywords: quantum information geometry, Bures metric, gradient flow, Lindblad dynamics, decoherence-free subspaces, many-body localization, Lyapunov functional

•May 12, 2026•0 citations

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We extend the Riemannian Bures gradient-flow analysis of the competing-poles potential Vdyn[ρ]V_{\mathrm{dyn}}[\rho]Vdyn​[ρ] from the d=2d=2d=2 qubit Bloch ball to the 8-dimensional manifold M∘(C3)\mathcal{M}_\circ(\mathbb{C}^3)M∘​(C3) of full-rank qutrit density matrices. Using the coordinate-free contravariant inverse Bures metric Gρ−1(H)=2(ρH+Hρ)−4Tr⁡(ρH)ρG_\rho^{-1}(H) = 2(\rho H + H\rho) - 4\operatorname{Tr}(\rho H)\rhoGρ−1​(H)=2(ρH+Hρ)−4Tr(ρH)ρ and the normalized SU(3)SU(3)SU(3) Hilbert–Schmidt purity distance BHS(ρ)=12(3Tr⁡(ρ2)−1)B_{\mathrm{HS}}(\rho) = \frac{1}{2}(3\operatorname{Tr}(\rho^2)-1)BHS​(ρ)=21​(3Tr(ρ2)−1), we prove the following analytical results without boundary clamping (ϵ=0\epsilon = 0ϵ=0): Operator Commutation: Every critical point on M∘(Cd)\mathcal{M}_\circ(\mathbb{C}^d)M∘​(Cd) commutes with the reference pure projector Π∗\Pi_*Π∗​. Orientation Reduction: At fixed spectrum, the non-logarithmic outward eigenvalue pull is strictly increasing in ρ00\sqrt{\rho_{00}}ρ00​​, reducing the SU(3)SU(3)SU(3) boundary-orientation problem to a two-parameter diagonal family. Forward Invariance: The one-sided derivative of λmin⁡(ρ)\lambda_{\min}(\rho)λmin​(ρ) along the flow is strictly positive on the analytic collar 0<λmin⁡≤6.84×10−50 < \lambda_{\min} \le 6.84 \times 10^{-5}0<λmin​≤6.84×10−5, establishing manifold forward invariance (with sharp numerical threshold λcollar=0.00481018\lambda_{\mathrm{collar}} = 0.00481018λcollar​=0.00481018). Transverse Uniqueness: No asymmetric critical points (λ2≠λ3\lambda_2 \neq \lambda_3λ2​=λ3​) exist despite the transverse potential's concavity. Coaxial Attractor: The coaxial balance equation admits a unique root p∗=0.961593764p^* = 0.961593764p∗=0.961593764, λ⊥∗=0.019203118\lambda_\perp^* = 0.019203118λ⊥∗​=0.019203118, BHS(ρ∗)=0.888100B_{\mathrm{HS}}(\rho^*) = 0.888100BHS​(ρ∗)=0.888100. Global Convergence: By LaSalle's Invariance Principle, every trajectory initialized in M∘(C3)\mathcal{M}_\circ(\mathbb{C}^3)M∘​(C3) converges globally to ρ∗\rho^*ρ∗. Includes complete standalone Python verification suite (ntcr_d3_verify.py) and rigorous numerical cross-checks across 105 complex starts.
Harmonic Maps on the Bloch Ball with the Bures Metric: Local Jets, the Global S3S^{3}S3 Spectrum, and the Distortion Observable D,QD, QD,Q
This repository contains the complete replication package for the research paper: "Harmonic Maps on the Bloch Ball with the Bures Metric: Local Jets, the Global S3S^3S3 Spectrum, and the Distortion Observable D,QD, QD,Q". We study harmonic self-maps Φ:M→M\Phi : \mathcal{M} \to \mathcal{M}Φ:M→M on the open single-qubit Bloch ball equipped with the Bures (quantum Fisher information) metric in the sense of Eells and Sampson. Key Contributions:Complete Affine Classification: We prove that a nondegenerate affine map is harmonic if and only if it is a global isometry (A∈O(3)A \in O(3)A∈O(3) and t=0t = 0t=0), recovering the exact boundary between unitary quantum channels (SO(3)SO(3)SO(3)) and positive but not completely positive (non-CP) maps. The Distortion Observable DDD: We introduce the Riemannian invariant D(Φ):=gΦ(r)(τ,τ)D(\Phi) := g_{\Phi(r)}(\tau, \tau)D(Φ):=gΦ(r)​(τ,τ) as a geometric distance to unitarity and evaluate it in closed form for the dephasing, depolarizing, and amplitude-damping channels. Pointwise Jacobi Jet (Block A): We compute the exact covariant closed form of the infinitesimal quadratic expansion QQQ around the identity map at every point of the ball, proving it descends canonically to the Bures metric on the effective quotient of generators. Global S3S^3S3 Spectral Geometry (Block B): By establishing a global isometry between the Bloch ball and an open hemisphere of the round 3-sphere of radius 1/21/21/2, we completely solve the global spectrum of the linearized tension (Jacobi) operator T=Δgrough+8 IdT = \Delta^{\text{rough}}_g + 8\,\text{Id}T=Δgrough​+8Id. Via representation theory (Schur's lemma applied to the SU(2)L×SU(2)RSU(2)_L \times SU(2)_RSU(2)L​×SU(2)R​ action), we determine the exact closed-form eigenvalues and multiplicities for both the gradient (Weitzenböck) and the toroidal sectors. Repository Contents (.zip file):1. Manuscript (PDF): The full text of the research article including complete derivations and theoretical proofs.2. Replication Source Code (Python/SymPy): Independently runnable scripts (relying only on `sympy`, `numpy`, and `mpmath`) that verify every closed-form identity, perform the high-precision finite-difference checks, and execute the numerical SVD diagonalization confirming the full closure of the toroidal sector at degrees ℓ=2\ell = 2ℓ=2 and ℓ=3\ell = 3ℓ=3. Keywords: harmonic maps; Bures metric; quantum Fisher information; Bloch ball; monotone metrics; Weitzenböck formula; homogeneous spaces; S3S^3S3 spectral geometry.
Version 2 (25 October 2026) adds an erratum. The main results of this record check out: the inverse Bures metric, the eigenvalue equation (6), the boundary collar 0 < lambda_min < 0.0564, both tables and the attractor p* = 0.9412018 were reproduced independently. Two steps in the proof text are corrected; the 0.0564 collar is labeled as established numerically (the collar proven analytically in version 3 of the Bures paper is 0.0261); and the weights are labeled as modeling choices, because the results depend on them. Every number in the erratum is printed by the attached script (addendum_d2_checks.py), included with its saved output. Version 1 remains available unchanged. The complete d=2 analysis, with full proofs, is at https://doi.org/10.5281/zenodo.23605523 Original description (version 1): We present an exact analytical and numerical analysis of the d=2 qubit reduction and logarithmic boundary barrier dynamics within the Nerik Theory of Conscious Refinement (NTCR V4.27). Through contravariant metric derivation and 50-digit numerical verification, we establish the exact Riemannian Bures eigenvalue velocity equation, superseding earlier heuristic coaxial approximations. By formulating the two-pole structured complexity C(rho) using the smooth Hilbert-Schmidt purity distance B_HS(rho) = 2 Tr((rho - I/2)^2) = r^2 alongside the Bures distance d_B^2(rho, \Pi_*) to the pure boundary projector \Pi_* = |\psi_*><\psi_*| and the von Neumann relative entropy barrier D(rho || I/2) = ln 2 - S(\rho), we eliminate the off-axis 1/sqrt(\lambda_{min}) boundary cusp and remove all ad-hoc coherence penalties and regularization parameters (\epsilon = 0). We prove analytically that: (i) radial Bloch vectors are exact eigenvectors of the 2x2 inverse Bures metric tensor G^{-1} with eigenvalue 1 - r^2 = 4 \lambda_{min} (1 - \lambda_{min}); (ii) d\lambda_{min}/d\tau > 0 holds uniformly across all angles \theta \in [0, 2\pi) inside the boundary collar 0 < \lambda_{min} < 0.0564; and (iii) no off-axis critical points exist anywhere in the full-rank manifold \mathcal{M}_\circ, driving all interior trajectories to the unique emergent interior attractor (p^* = 0.941202, x_c^* = 0, \lambda_{min}^* = 0.058798).
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.
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 formulate a geometric extension of open quantum dynamics in which the density matrix evolves as a Riemannian gradient flow on the Bures manifold. The evolution equation \frac{d\rho}{dt} = -i[H,\rho] - \lambda \nabla_B \Phi[\rho] drives the state toward regions of minimal information degradation, defined by the functional \Phi[\rho], the instantaneous rate of relative-entropy change under the bare Lindbladian. This deterministic flow preserves trace and positivity, and introduces a single new universal constant \lambda>0 while recovering standard Lindblad dynamics as \lambda\to0. Under a Unitary Orthogonality Condition – exactly satisfied for pure dephasing and approximately in many-body localized phases – \Phi becomes a Lyapunov functional, and its critical points (decoherence-free subspaces) are asymptotically stable attractors. Applied to disordered spin chains, the framework reproduces the known entropy‑suppression scaling \Phi \lesssim \gamma N (J/W)^2. The theory offers a falsifiable, information‑geometric principle where quantum states intrinsically minimise their own information loss. Keywords: quantum information geometry, Bures metric, gradient flow, Lindblad dynamics, decoherence-free subspaces, many-body localization, Lyapunov functional
The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information. Riemannian gradient descent (RGD) at unit step size -- the fixed-point iteration used in practice -- converges rapidly, yet existing analyses present a dichotomy: unit-step guarantees carry worst-case exponential dependence on the dimension, while dimension-independent guarantees require small step sizes that forfeit the empirical speed. We resolve this dichotomy, not by improving the guarantees for unit-step RGD, but by proposing a Projected RGD algorithm that achieves dimension-independent linear convergence at unit step size. The achieved rate, (1−κ−3/2)(1 - κ^{-3/2})(1−κ−3/2), where κκκ is the condition number of the ensemble, also polynomially improves on the best small-step guarantee (κ3/2κ^{3/2}κ3/2 versus κ5/2κ^{5/2}κ5/2 iteration complexity). The crux is a novel Projection Lemma: clipping the eigenvalues of a positive matrix to an interval [α,β][α, β][α,β] is the closed-form, non-expansive (1-Lipschitz) BW-metric projection onto the set {S:αI≤S≤βI}\{S : αI \leq S \leq βI\}{S:αI≤S≤βI} -- a statement which, unlike its known one-sided counterpart, does not follow from convexity. The projection is moreover free: it reuses an eigendecomposition the next iteration must perform in any case, so the projected and unprojected iterations cost the same per step. The same analysis covers the invariant matrix projection problem of Brahmachari et al. (2025), whose fixed-point algorithm we identify as unit-step RGD on a totally geodesic submanifold, thereby extending the dimension-independent guarantee to that setting verbatim.