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

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 $S^3$ 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.