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.
Abstract We construct spectral triples of one- and two-qubit states using the Hilbert-Schmidt operatorial formulation, and study the Connes spectral distances. We also construct the Dirac operator corresponding to the normal quantum trace distances. Based on the Connes spectral distances, we propose some definitions of quantum discord and coherence measure of quantum states, and explicitly calculate the coherence of one-qubit states. We also study some simple cases about two-qubit states, and the corresponding spectral distances satisfy the Pythagoras theorem. These results are significant for studies on physical relations and geometric structures of qubits and other quantum states.
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.
In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces. We prove that there are some finite spectral triples in which the Lipschitz seminorms are equal to the operator norms. We also explicitly construct some spectral triples in which the Connes spectral distances between quantum states are exactly the quantum trace distances. These results are helpful for us to better study the relationships among Connes spectral distance, quantum trace distance and other quantum distance measures. These concrete examples are significant for studies of geometric structures of finite spectral triples and mathematical relations of qubits and other quantum states in the framework of noncommutative geometry.
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.
When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.