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 $Ψ=Φ^2$ 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 $6/(1-4t)$ and extinction time $T=1/4$. Thus the metric remains inside the monotone cone for all $t<T$ 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 $Δ_{\mathbb S^3}+3$, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.
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=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) and extinction time T=1/4. Thus the metric remains inside the monotone cone for all t<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+3, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap 5 after removal of the scaling mode.
Harmonic Maps on the Bloch Ball with the Bures Metric: Local Jets, the Global S3 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 S3 Spectrum, and the Distortion Observable D,Q". We study harmonic self-maps 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) 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(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 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 S3 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=Δgrough+8Id. Via representation theory (Schur's lemma applied to the SU(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 and ℓ=3. Keywords: harmonic maps; Bures metric; quantum Fisher information; Bloch ball; monotone metrics; Weitzenböck formula; homogeneous spaces; S3 spectral geometry.
It is well known that quantum states, particularly qubits and BECs qubits, can be represented by a spherical structure known as the Bloch sphere. Several error models describe errors that effectively shrink the Bloch sphere, such as the depolarizing channel and particle loss, which can be mitigated through redundancy error correction models. In contrast, it may be possible to realize error correction by directly deforming the Bloch sphere itself. Geometric flows, particularly the Ricci flow, describe the evolution of a metric over time. In the Ricci flow equation, the metric contracts along directions of higher curvature, leading to a uniformization of curvature across the manifold. In our approach, we utilize such geometric flows to achieve error correction by allowing the metric of the Bloch sphere naturally defined by geometric quantities such as the Fubini-Study or Bures metric to evolve. To employ these geometric quantities and equations, it is necessary to endow the Bloch sphere with a differentiable manifold structure, which can indeed be done. Moreover, our error correction scheme is distinctive in that it does not rely on redundancy.
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.
In a recent work we introduced a finite scalar-qubit-scalar spectral triple whose Connes distance reproduces the Bloch-ball trace metric up to a factor 2/Λ, where Λ2=μL2+μR2 for isotropic link strengths μL,μR appearing in the proposed Dirac operator. Here we analyse time-dependent variation of these links, independent of the qubit dynamics. In doing so, we separate the distinguishability and calibration contributions to the spectral rate. A depolarizing example demonstrates spectral growth despite a monotonic contraction of the trace-distance. More generally, we characterize the admissible trace-distance histories compatible with a given positive spectral record and exhibit contracting and reviving realizations. At fixed Λ, we also study a sinusoidal link imbalance that preserves the Dirac spectrum and qubit metric, while the scalar-scalar distance oscillates with half the period of the imbalance.
We extend the Riemannian Bures gradient-flow analysis of the competing-poles potential Vdyn[ρ] from the d=2 qubit Bloch ball to the 8-dimensional manifold M∘(C3) of full-rank qutrit density matrices. Using the coordinate-free contravariant inverse Bures metric Gρ−1(H)=2(ρH+Hρ)−4Tr(ρH)ρ and the normalized SU(3) Hilbert–Schmidt purity distance BHS(ρ)=21(3Tr(ρ2)−1), we prove the following analytical results without boundary clamping (ϵ=0): Operator Commutation: Every critical point on M∘(Cd) commutes with the reference pure projector Π∗. Orientation Reduction: At fixed spectrum, the non-logarithmic outward eigenvalue pull is strictly increasing in ρ00, reducing the SU(3) boundary-orientation problem to a two-parameter diagonal family. Forward Invariance: The one-sided derivative of λmin(ρ) along the flow is strictly positive on the analytic collar 0<λmin≤6.84×10−5, establishing manifold forward invariance (with sharp numerical threshold λcollar=0.00481018). Transverse Uniqueness: No asymmetric critical points (λ2=λ3) exist despite the transverse potential's concavity. Coaxial Attractor: The coaxial balance equation admits a unique root p∗=0.961593764, λ⊥∗=0.019203118, BHS(ρ∗)=0.888100. Global Convergence: By LaSalle's Invariance Principle, every trajectory initialized in M∘(C3) converges globally to ρ∗. Includes complete standalone Python verification suite (ntcr_d3_verify.py) and rigorous numerical cross-checks across 105 complex starts.
We show that the equal-prior Helstrom trace-distance geometry of qubit states is recovered from Connes spectral distance in a finite scalar-qubit-scalar model. The two scalar reference sectors couple isotropically to the qubit block through identity Dirac links, so that the full Bloch ball, including mixed states, inherits its standard chordal trace-distance geometry from the finite spectral metric. The scalar-sector distances serve a distinct calibration role: they determine the individual link lengths, satisfy a Pythagorean consistency relation, and reconstruct the middle-sector scale.