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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 \(D, Q\)

Federico Kasjan

Published Aug 24, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.22073113

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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.
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.
Three results. (1) We prove the tensor-level conformal identity dsBK2=4γ2(r) dsBures2 between the Bures metric and the Beltrami–Klein metric on the open qubit Bloch ball, where γ2(r)=1/(1−r2) is the squared Lorentz factor. (2) For the visibility coordinate V=2p−1 of a binary quantum measurement, the classical Bernoulli Fisher information takes the closed form I(V)=1/(1−V2)=γ2(V). (3) The conformal structure is special to qubits; for N≥3, the Bures metric on full-rank density matrices has a non-constant sectional curvature at the maximally mixed state and hence a non-vanishing Weyl tensor and no conformal equivalence to any constant-curvature hyperbolic model (Theorem 2). We outline an experimentally testable operational consequence where sequential non-collinear weak measurements on a qubit predict a Thomas–Wigner rotation with a closed-form purity dependence that deviates from the Pancharatnam baseline at intermediate visibility.
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.