Geometric Response and State-Resolved Decoherence: Microscopic Born–Markov Derivation for a Commuting Qubit–Bath Coupling

We analyze the angular dependence of the purity-loss rate in a restricted class of open quantum systems: a qubit coupled to a thermal bosonic bath through a commuting interaction operator. Within the Born–Markov approximation, we derive the dissipative generator in closed form andestablish four results. First, the restricted induced trace norm of the generator (on the Hermitian traceless subspace) isindependent of the coupling angle, ‖ℒ(𝐷)𝜃 ‖1→1,ℬ0 = 4𝜋𝜂𝑘𝐵 𝑇,and we give a closed-form proof of this fact based on the rotational structure of the coupling family, together with an independent numerical cross-check.Second, the initial purity-loss rate of a pure reference state depends on the coupling angle through the geometric quantity Γ(𝜃) = |⟨𝜓|𝐴(𝜃)|𝜓⟩|2, which for the qubit family 𝐴(𝜃) = cos 𝜃 𝜎𝑥 + sin 𝜃 𝜎𝑧 takes the exact form Γ(𝜃) = (𝑒𝑥 cos 𝜃 + 𝑒𝑧 sin 𝜃)2. Third, the local curvature of Γ(𝜃) at its maximum defines a curvature-derived angular scale 𝜎geo = 1/√2, universal within this coupling family and independent of every physical parameter of the model; the two forms agree to within ∼ 5% at |𝜃 − 𝜇| = 𝜎geo (Appendix E). Fourth, we quantify explicitly the algebraic (non-exponential) decay of the Markov tail integral for the Ohmic–Lorentzian bath considered here, and we show that a slowly time-dependent coupling angle, 𝜃(𝑡), admits a well-defined adiabatic-Markovian generator; we then prove that the resulting purity-loss functional is not proportional, in general, to the trajectory-integrated Frobenius-norm witness of Ref. [1], identifying the specific obstructions (coherent-precession contamination and a linear-versus-quadratic mismatch in Bloch-vector language) responsible, and we pose as an open problem the narrower question of whether any restricted or reformulated version of the witness admits a well-defined relation to the present results. These results provide an exact microscopic benchmark for trajectory-dependent effective decoherence models: the angular dependence originates from the geometry of the system–environment coupling and the evolving state, not from an intrinsic angular preference of the bath, while a full reduction of the phenomenological model of Ref. [1] to this microscopic setting remains open.