On a generalized Central Limit Theorem and Large Deviations for Homogeneous Open Quantum Walks

The paper proves a generalized CLT for homogeneous open quantum random walks (HOQRWs): the rescaled position converges, in Fortet–Mourier distance, to a mixture of Gaussians with weights given by absorption probabilities into invariant domains (enclosures) of the local channel; see Theorem 1.1 and Theorem 4.4, with the measure decomposition via absorption operators in Lemma 4.3 and the single-enclosure CLT in Theorem 3.5 . The candidate solution follows the same structure: (i) decompose the fast recurrent part into enclosures, (ii) define absorption-based weights a_α(ρ), (iii) condition to a minimal enclosure to get a single-Gaussian CLT via spectral deformation and Bryc’s theorem, and (iv) take convex mixtures with the bounded-Lipschitz metric to conclude. Two minor issues in the model’s write-up are: (1) the Doob transform is described using a deterministic martingale M_n = Tr(L^{*n}(p_V)ρ_0), whereas the paper correctly uses the random martingale Y_n = Tr(A(V)ρ_n) to define the change of measure dP′/dP = Y_n/E[Y_0] (Lemma 3.1) ; and (2) the claim that the coin remains in V after a finite time is stronger than what is proved (the paper shows asymptotic absorption in the sense lim‖p_Vρ_np_V − ρ_n‖ = 0 under the changed measure) . Aside from these corrections, the model’s argument matches the paper’s approach, including the independence of the Gaussian parameters from the choice of minimal enclosure inside each χα (Lemma 4.2) and the final Fortet–Mourier distance estimate (Theorem 4.4) .