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arXiv:2610.01844

Ballistic Speed and Potential-Theoretic Recurrence for Homogeneous Open Quantum Random Walks

Ameur Dhahri, Chul Ki Ko, Farrukh Mukhamedov, Hyun Jae Yoo

math.FAmath-phmath.OAmath.PRquant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

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44 pages

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Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic (h)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.
Abstract We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.
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Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.