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Quantum Doeblin Coefficients: Interpretations and Applications

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Quantum Doeblin Coefficients: Interpretations and Applications

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AbstractIn classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, on the fairness of noisy quantum models, and on mixing, indistinguishability, and decoupling times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a quantum channel. Furthermore, in all of these applications, our analysis using quantum Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.► BibTeX data@article{George2026quantumdoeblin, doi = {10.22331/q-2026-05-22-2115}, url = {https://doi.org/10.22331/q-2026-05-22-2115}, title = {Quantum {D}oeblin {C}oefficients: {I}nterpretations and {A}pplications}, author = {George, Ian and Hirche, Christoph and Nuradha, Theshani and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2115}, month = may, year = {2026} }► References [1] W. Doeblin. ``Sur les proprietes asymptotiques de mouvement régis par certains types de chaines simples''. Bulletin mathematique de la Societe Roumaine des Sciences 39, 57–115 (1937). url: http://www.jstor.org/stable/43769809. http://www.jstor.org/stable/43769809 [2] Anuran Makur and Japneet Singh. ``Doeblin coefficients and related measures''. IEEE Transactions on Information Theory 70, 4667–4692 (2024). https://doi.org/10.1109/TIT.2024.3367856 [3] Hao Chen, Jiacheng Tang, and Abhishek Gupta. ``Change detection of Markov kernels with unknown pre and post change kernel''. In 2022 IEEE 61st Conference on Decision and Control (CDC). Pages 4814–4820. (2022). https://doi.org/10.1109/CDC51059.2022.9992982 [4] Vrettos Moulos. ``Finite-time analysis of round-robin Kullback-Leibler upper confidence bounds for optimal adaptive allocation with multiple plays and Markovian rewards''. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems. Volume 33, pages 7863–7874. Curran Associates, Inc. (2020). url: https://proceedings.neurips.cc/paper_files/paper/2020/file/597c7b407a02cc0a92167e7a371eca25-Paper.pdf. https://proceedings.neurips.cc/paper_files/paper/2020/file/597c7b407a02cc0a92167e7a371eca25-Paper.pdf [5] Jeffrey S. Rosenthal. ``Minorization conditions and convergence rates for Markov chain Monte Carlo''. Journal of the American Statistical Association 90, 558–566 (1995). https://doi.org/10.2307/2291067 [6] Jeffrey M. Alden and Robert L. Smith. ``Rolling horizon procedures in nonhomogeneous Markov decision processes''. Operations Research 40, S183–S194 (1992). https://doi.org/10.1287/opre.40.3.S183 [7] Jacob Steinhardt and Percy Liang. ``Learning fast-mixing models for structured prediction''.

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[6] Abdessatar Souissi, "Ergodic Theory of Inhomogeneous Quantum Processes", arXiv:2506.12280, (2025). [7] Theshani Nuradha, Ian George, and Christoph Hirche, "Non-Linear Strong Data-Processing for Quantum Hockey-Stick Divergences", arXiv:2512.16778, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-22 12:47:12). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-22 12:47:11: Could not fetch cited-by data for 10.22331/q-2026-05-22-2115 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractIn classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, on the fairness of noisy quantum models, and on mixing, indistinguishability, and decoupling times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a quantum channel. Furthermore, in all of these applications, our analysis using quantum Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.► BibTeX data@article{George2026quantumdoeblin, doi = {10.22331/q-2026-05-22-2115}, url = {https://doi.org/10.22331/q-2026-05-22-2115}, title = {Quantum {D}oeblin {C}oefficients: {I}nterpretations and {A}pplications}, author = {George, Ian and Hirche, Christoph and Nuradha, Theshani and Wilde, Mark M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2115}, month = may, year = {2026} }► References [1] W. Doeblin. ``Sur les proprietes asymptotiques de mouvement régis par certains types de chaines simples''. 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Quantum Doeblin Coefficients: Interpretations and Applications

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