The Geometry of Qubit Decoherence: Linear vs. Nonlinear Dynamics in the Bloch Ball

TL;DR

The paper investigates the geometry of qubit decoherence under GKSL dynamics by presenting two complementary descriptions: a standard linear Bloch-ball analysis and a nonlinear, SU(2)–symmetric formulation that decouples radial dissipation from angular orientation. It derives explicit isotropic and anisotropic solutions, identifies trajectory invariants, and analyzes fixed points, including a Liouvillian exceptional point that governs the transition between oscillatory and overdamped regimes. The SU(2) perspective expresses the state as $\rho_{r,\theta,\phi}=\tfrac{1}{2}(I+r\,\sigma_{\theta,\phi})$ and yields Ricatti-like equations for orientation, highlighting a natural generalization to open qudits via the coset $SU(N)/U(1)^{N-1}$ and the radial/angular separation. Together, these viewpoints provide geometric, design-oriented insights for stabilizing qubits and formulating a framework for open quantum systems, with potential applications in decoherence-free subspaces and quantum information storage.

Abstract

We present two complementary approaches to the GKSL equation for an open qubit. The first, based on linearity, yields solutions illustrated by mixed states trajectories in the Bloch ball, including non-random asymptotic fixed points, and exceptional points. The second, exploiting the SU(2) symmetry, leads to a nonlinear dynamical system that separates angular dynamics from radial dissipation. This symmetry-based perspective offers a promising route toward generalisation to open qudits.

Figures (6)

• Figure 1: Trajectory of the quantum state given by Equations \ref{['eq: Initial100']}\ref{['eq: Initial101']} related to the initial condition (left) $\vec{a}(0)=(1,0,0)^T$ and (right) $\vec{a}(0)=(1/\sqrt{2})(1,0,1)^T$ , with the Hamiltonian $H=\omega_0\sigma_z$, where $\omega_0=10$, $h=1$.
• Figure 2: Evolution of the components $a_i(t)$ of the example related to equation \ref{['eq: Initial101']}, related to the initial condition $\vec{a}(0)=(1/\sqrt{2})(1,0,1)$, with environment isotropic interactions
• Figure 3: Trajectory of the system in the Bloch ball, for the three regimes (left) oscillation/sub-critical $\beta<1$, (center) critical $\beta=1$, and super-critical $\beta>1$. In the three examples, the contribution of the Hamiltonian is set to $\omega_0=10$, while the decay rates are (left) $h=2$, (center) $h=20$, and (right) $h=50$. The initial condition is the same for the three cases $\vec{a}(0)=(1,0,0)^T$.
• Figure 4: Decay comparison of the examples in Fig. \ref{['fig: competition']}, for the three regimes oscillation/sub-critical $\beta<1$, critical $\beta=1$, and super-critical $\beta>1$. The system related to the critical case decays faster then the others.
• Figure 5: Plot of the evolution of the components $a_i(t)$ of the $\vec{a}(t)$ associated to the Pauli matrices $\sigma_i$ with $i=1,2,3$, for the (top) subcritical, (center) critical, and (bottom) supercritical regimes.