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arXiv:2604.09422

Periodicity in Ergodic Quantum Processes

Owen Ekblad, Jeffrey Schenker

math-phmath.DSmath.FAquant-phSubmitted Apr 10, 2026• Announced Apr 13, 2026
v1 · Apr 10, 2026v2 · Aug 26, 2026
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Abstract

We study the periodic properties of sequences of quantum channels sampled from an ergodic stochastic process satisfying a natural irreducibility condition. We relate these periodic properties to certain global spectral data defined by the sequence of quantum channels, proving a general Perron-Frobenius-type theorem. We give examples to motivate the theory and conclude with some open problems and conjectures.

Comments
36 pages; supported by National Science Foundation Grant No. 2153946

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We study the periodic properties of sequences of quantum channels sampled from an ergodic stochastic process satisfying a natural irreducibility condition. We relate these periodic properties to certain global spectral data defined by the sequence of quantum channels, proving a general Perron-Frobenius-type theorem. We give examples to motivate the theory and conclude with some open problems and conjectures.
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Any quantum process is represented by a sequence of quantum channels. We consider ergodic processes, obtained by sampling channel valued random variables along the trajectories of an ergodic dynamical system. Examples of such processes include the effect of repeated application of a fixed quantum channel perturbed by arbitrary correlated noise, or a sequence of channels drawn independently and identically from an ensemble. Under natural irreducibility conditions, we obtain a theorem showing that the state of a system evolving by such a process converges exponentially fast to an ergodic sequence of states depending on the process, but independent of the initial state of the system. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk. Further applications and physical implications of our results are discussed in the companion paper [11].
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