Self-restricting Noise and Exponential Relative Entropy Decay Under Unital Quantum Markov Semigroups
Summarize this article with:
AbstractStates of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.Featured image: Relative entropy of one qubit within a two-qubit system to its completely mixed fixed point at a fixed time. The two qubits interact via the Hamiltonian as in Example 4.2 of the paper, but only the other qubit is exposed to dissipative noise. As the decay rate induced by that noise increases, the relative entropy shown dips, then rebounds.Popular summaryIn the absence of error correction or perfect isolation, a quantum system's exposure to its environment typically induces noise. It is known that when a system's dynamics arise entirely from noisy environmental interactions, its entropy approaches that of equilibrium at an exponential rate. However, many of these known results do not immediately apply when combining noise with internal processes. Herein, theories of exponential equilibration are extended to a wide class of such combined processes. When noise is restricted to a part of the system, it might still be spread by the system's internal dynamics. In such scenarios, extremely rapid noisy interactions actually reduce the system's decay rate toward equilibrium. This 'self-restriction' occurs because the noise process suppresses interactions between the highly noised part and the rest of the system.► BibTeX data@article{LaRacuente2026selfrestricting, doi = {10.22331/q-2026-03-04-2010}, url = {https://doi.org/10.22331/q-2026-03-04-2010}, title = {Self-restricting {N}oise and {E}xponential {R}elative {E}ntropy {D}ecay {U}nder {U}nital {Q}uantum {M}arkov {S}emigroups}, author = {LaRacuente, Nicholas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2010}, month = mar, year = {2026} }► References [1] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf. Introduction to quantum noise, measurement, and amplification. Reviews of Modern Physics, 82 (2): 1155–1208, April 2010. 10.1103/RevModPhys.82.1155. URL https://doi.org/10.1103/RevModPhys.82.1155. Publisher: American Physical Society. https://doi.org/10.1103/RevModPhys.82.1155 [2] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/q-2018-08-06-79. URL https://quantum-journal.org/papers/q-2018-08-06-79/. https://doi.org/10.22331/q-2018-08-06-79 https://quantum-journal.org/papers/q-2018-08-06-79/ [3] Mark M. Wilde.
Quantum Information Theory.
Cambridge University Press, Cambridge, 2 edition, 2017. ISBN 978-1-107-17616-4. 10.1017/9781316809976. URL https://www.cambridge.org/core/books/quantum-information-theory/247A740E156416531AA8CB97DFDAE438. https://doi.org/10.1017/9781316809976 https://www.cambridge.org/core/books/quantum-information-theory/247A740E156416531AA8CB97DFDAE438 [4] R Rebolledo. Decoherence of quantum Markov semigroups. Annales de l'Institut Henri Poincare (B) Probability and Statistics, 41 (3): 349–373, May 2005. ISSN 02460203. 10.1016/j.anihpb.2004.12.003. URL https://linkinghub.elsevier.com/retrieve/pii/S0246020305000257. https://doi.org/10.1016/j.anihpb.2004.12.003 https://linkinghub.elsevier.com/retrieve/pii/S0246020305000257 [5] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. Rapid Thermalization of Spin Chain Commuting Hamiltonians.
Physical Review Letters, 130 (6): 060401, February 2023. 10.1103/PhysRevLett.130.060401. URL https://doi.org/10.1103/PhysRevLett.130.060401. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.130.060401 [6] Andreas Bluhm, Ángela Capel, and Antonio Pérez-Hernández. Exponential decay of mutual information for Gibbs states of local Hamiltonians. Quantum, 6: 650, February 2022. 10.22331/q-2022-02-10-650. URL https://quantum-journal.org/papers/q-2022-02-10-650/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-02-10-650 https://quantum-journal.org/papers/q-2022-02-10-650/ [7] G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48 (2): 119–130, June 1976. ISSN 0010-3616, 1432-0916. 10.1007/BF01608499. URL http://link.springer.com/10.1007/BF01608499. https://doi.org/10.1007/BF01608499 [8] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. Journal of Mathematical Physics, 17 (5): 821–825, 1976. ISSN 0022-2488. 10.1063/1.522979. URL https://doi.org/10.1063/1.522979. https://doi.org/10.1063/1.522979 [9] Hisaharu Umegaki. Conditional expectation in an operator algebra. IV. Entropy and information.
Kodai Mathematical Seminar Reports, 14 (2): 59–85, 1962. ISSN 0023-2599. 10.2996/kmj/1138844604. URL https://projecteuclid.org/euclid.kmj/1138844604. https://doi.org/10.2996/kmj/1138844604 https://projecteuclid.org/euclid.kmj/1138844604 [10] Li Gao and Cambyse Rouzé.
Complete Entropic Inequalities for Quantum Markov Chains. Archive for Rational Mechanics and Analysis, 245 (1): 183–238, July 2022. ISSN 1432-0673. 10.1007/s00205-022-01785-1. URL https://doi.org/10.1007/s00205-022-01785-1. https://doi.org/10.1007/s00205-022-01785-1 [11] Li Gao, Marius Junge, and Haojian Li.
Geometric Approach Towards Complete Logarithmic Sobolev Inequalities. February 2021. 10.48550/arXiv.2102.04434. URL http://arxiv.org/abs/2102.04434. arXiv:2102.04434 [quant-ph]. https://doi.org/10.48550/arXiv.2102.04434 arXiv:2102.04434 [12] Marius Junge, Nicholas Laracuente, and Cambyse Rouzé. Stability of Logarithmic Sobolev Inequalities Under a Noncommutative Change of Measure. Journal of Statistical Physics, 190 (2): 30, December 2022. ISSN 1572-9613. 10.1007/s10955-022-03026-x. URL https://doi.org/10.1007/s10955-022-03026-x. https://doi.org/10.1007/s10955-022-03026-x [13] Li Gao, Marius Junge, and Nicholas LaRacuente. Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions.
Annales Henri Poincaré, 21 (11): 3409–3478, November 2020a. ISSN 1424-0661. 10.1007/s00023-020-00947-9. URL https://doi.org/10.1007/s00023-020-00947-9. https://doi.org/10.1007/s00023-020-00947-9 [14] Michael J. Kastoryano and Kristan Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. Journal of Mathematical Physics, 54 (5): 052202, May 2013. ISSN 0022-2488. 10.1063/1.4804995. URL https://doi.org/10.1063/1.4804995. https://doi.org/10.1063/1.4804995 [15] Ameur Dhahri, Franco Fagnola, and Rolando Rebolledo. The Decoherence-Free Subalgebra of a Quantum Markov Semigroup with Unbounded Generator.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 13 (03): 413–433, September 2010. ISSN 0219-0257, 1793-6306. 10.1142/S0219025710004176. URL https://www.worldscientific.com/doi/abs/10.1142/S0219025710004176. https://doi.org/10.1142/S0219025710004176 [16] Franco Fagnola, Emanuela Sasso, and Veronica Umanità. The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups. Journal of Mathematical Physics, 60 (7): 072703, July 2019. ISSN 0022-2488, 1089-7658. 10.1063/1.5030954. URL https://pubs.aip.org/jmp/article/60/7/072703/319785/The-role-of-the-atomic-decoherence-free-subalgebra. https://doi.org/10.1063/1.5030954 https://pubs.aip.org/jmp/article/60/7/072703/319785/The-role-of-the-atomic-decoherence-free-subalgebra [17] Vladislav Popkov, Simon Essink, Carlo Presilla, and Gunter Schütz. Effective quantum Zeno dynamics in dissipative quantum systems. Physical Review A, 98 (5): 052110, November 2018. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.98.052110. URL https://doi.org/10.1103/PhysRevA.98.052110. https://doi.org/10.1103/PhysRevA.98.052110 [18] Li Gao, Marius Junge, and Nicholas LaRacuente. Relative entropy for von Neumann subalgebras. International Journal of Mathematics, 31 (06): 2050046, June 2020b. ISSN 0129-167X. 10.1142/S0129167X20500469. URL https://www.worldscientific.com/doi/abs/10.1142/S0129167X20500469. Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0129167X20500469 [19] Mihai Pimsner and Sorin Popa. Entropy and index for subfactors. Annales scientifiques de l'École Normale Supérieure, 19 (1): 57–106, 1986. ISSN 1873-2151. 10.24033/asens.1504. URL http://www.numdam.org/item/?id=ASENS_1986_4_19_1_57_0. https://doi.org/10.24033/asens.1504 http://www.numdam.org/item/?id=ASENS_1986_4_19_1_57_0 [20] V. F. R. Jones. Index for subfactors. Inventiones mathematicae, 72 (1): 1–25, February 1983. ISSN 1432-1297. 10.1007/BF01389127. URL https://doi.org/10.1007/BF01389127. https://doi.org/10.1007/BF01389127 [21] Göran Lindblad. Completely positive maps and entropy inequalities. Communications in Mathematical Physics, 40 (2): 147–151, June 1975. ISSN 1432-0916. 10.1007/BF01609396. URL https://doi.org/10.1007/BF01609396. https://doi.org/10.1007/BF01609396 [22] Andreas Winter.
Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints. Communications in Mathematical Physics, 347 (1): 291–313, October 2016. ISSN 1432-0916. 10.1007/s00220-016-2609-8. URL https://doi.org/10.1007/s00220-016-2609-8. https://doi.org/10.1007/s00220-016-2609-8 [23] Andreas Bluhm, Ángela Capel, Paul Gondolf, and Antonio Pérez-Hernández. Continuity of Quantum Entropic Quantities via Almost Convexity. IEEE Transactions on Information Theory, 69 (9): 5869–5901, September 2023. ISSN 0018-9448, 1557-9654. 10.1109/TIT.2023.3277892. URL https://ieeexplore.ieee.org/document/10129917/. https://doi.org/10.1109/TIT.2023.3277892 https://ieeexplore.ieee.org/document/10129917/ [24] Andreas Bluhm, Ángela Capel, Paul Gondolf, and Antonio Pérez-Hernández. Corrections to “Continuity of Quantum Entropic Quantities via Almost Convexity”. IEEE Transactions on Information Theory, 70 (7): 5410–5411, July 2024. ISSN 0018-9448, 1557-9654. 10.1109/TIT.2024.3390839. URL https://ieeexplore.ieee.org/document/10504886/. https://doi.org/10.1109/TIT.2024.3390839 https://ieeexplore.ieee.org/document/10504886/ [25] J. Lin. Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37 (1): 145–151, January 1991. ISSN 00189448. 10.1109/18.61115. URL http://ieeexplore.ieee.org/document/61115/. https://doi.org/10.1109/18.61115 http://ieeexplore.ieee.org/document/61115/ [26] M. B. Hastings. Superadditivity of communication capacity using entangled inputs. Nature Physics, 5 (4): 255–257, April 2009. ISSN 1745-2481. 10.1038/nphys1224. URL https://www.nature.com/articles/nphys1224. Number: 4 Publisher: Nature Publishing Group. https://doi.org/10.1038/nphys1224 https://www.nature.com/articles/nphys1224 [27] Dénes Petz. Sufficiency of Channels Over Von Neumann Algebras.
The Quarterly Journal of Mathematics, 39 (1): 97–108, March 1988. ISSN 0033-5606. 10.1093/qmath/39.1.97. URL https://doi.org/10.1093/qmath/39.1.97. https://doi.org/10.1093/qmath/39.1.97 [28] Maximilian Schlosshauer. Quantum decoherence. Physics Reports, 831: 1–57, October 2019. ISSN 0370-1573. 10.1016/j.physrep.2019.10.001. URL https://www.sciencedirect.com/science/article/pii/S0370157319303084. https://doi.org/10.1016/j.physrep.2019.10.001 https://www.sciencedirect.com/science/article/pii/S0370157319303084 [29] Dieter Suter and Gonzalo A. Álvarez. Colloquium: Protecting quantum information against environmental noise. Reviews of Modern Physics, 88 (4): 041001, October 2016. 10.1103/RevModPhys.88.041001. URL https://doi.org/10.1103/RevModPhys.88.041001. Publisher: American Physical Society. https://doi.org/10.1103/RevModPhys.88.041001 [30] M. Beau, J. Kiukas, I. L. Egusquiza, and A. del Campo.
Nonexponential Quantum Decay under Environmental Decoherence.
Physical Review Letters, 119 (13): 130401, September 2017. 10.1103/PhysRevLett.119.130401. URL https://doi.org/10.1103/PhysRevLett.119.130401. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.119.130401 [31] I.A. Aloisio, G.A.L. White, C.D. Hill, and K. Modi. Sampling Complexity of Open Quantum Systems. PRX Quantum, 4 (2): 020310, April 2023. 10.1103/PRXQuantum.4.020310. URL https://doi.org/10.1103/PRXQuantum.4.020310. Publisher: American Physical Society. https://doi.org/10.1103/PRXQuantum.4.020310 [32] Patrick Hayden and Jonathan Sorce. A canonical Hamiltonian for open quantum systems. Journal of Physics A: Mathematical and Theoretical, 55 (22): 225302, May 2022. ISSN 1751-8121. 10.1088/1751-8121/ac65c2. URL https://dx.doi.org/10.1088/1751-8121/ac65c2. Publisher: IOP Publishing. https://doi.org/10.1088/1751-8121/ac65c2 [33] Marco Merkli. Dynamics of Open Quantum Systems II, Markovian Approximation. Quantum, 6: 616, January 2022. 10.22331/q-2022-01-03-616. URL https://quantum-journal.org/papers/q-2022-01-03-616/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-01-03-616 https://quantum-journal.org/papers/q-2022-01-03-616/ [34] Anton Arnold, Peter Markowich, Giuseppe Toscani, and Andreas Unterreiter. On logarithmic Sobolev inequalities, Csiszar-Kullback inequalities, and the rate of convergence to equilibrium for Fokker-Planck type equations. Preprint-Reihe Mathematik, 592: 77, 1998. [35] Sergey Bobkov and Prasad Tetali. Modified log-sobolev inequalities, mixing and hypercontractivity. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, STOC '03, pages 287–296, New York, NY, USA, June 2003. Association for Computing Machinery. ISBN 978-1-58113-674-6. 10.1145/780542.780586. URL https://dl.acm.org/doi/10.1145/780542.780586. https://doi.org/10.1145/780542.780586 [36] Leonard Gross.
Logarithmic Sobolev Inequalities. American Journal of Mathematics, 97 (4): 1061–1083, 1975a. ISSN 0002-9327. 10.2307/2373688. URL https://www.jstor.org/stable/2373688. Publisher: Johns Hopkins University Press. https://doi.org/10.2307/2373688 https://www.jstor.org/stable/2373688 [37] Leonard Gross. Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form.
Duke Mathematical Journal, 42 (3): 383–396, September 1975b. ISSN 0012-7094, 1547-7398. 10.1215/S0012-7094-75-04237-4. URL https://projecteuclid.org/journals/duke-mathematical-journal/volume-42/issue-3/Hypercontractivity-and-logarithmic-Sobolev-inequalities-for-the-Clifford-Dirichlet-form/10.1215/S0012-7094-75-04237-4.full. https://doi.org/10.1215/S0012-7094-75-04237-4 [38] Ivan Bardet and Cambyse Rouzé. Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates.
Annales Henri Poincaré, 23 (11): 3839–3903, November 2022. ISSN 1424-0661. 10.1007/s00023-022-01196-8. URL https://doi.org/10.1007/s00023-022-01196-8. https://doi.org/10.1007/s00023-022-01196-8 [39] Ivan Bardet. Estimating the decoherence time using non-commutative Functional Inequalities. October 2017. 10.48550/arXiv.1710.01039. URL http://arxiv.org/abs/1710.01039. arXiv:1710.01039 [math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.1710.01039 arXiv:1710.01039 [40] Ivan Bardet, Marius Junge, Nicholas Laracuente, Cambyse Rouzé, and Daniel Stilck França.
Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups. IEEE Transactions on Information Theory, 67 (5): 2878–2909, May 2021. ISSN 1557-9654. 10.1109/TIT.2021.3065452. Conference Name: IEEE Transactions on Information Theory. https://doi.org/10.1109/TIT.2021.3065452 [41] Daniel Stilck França and Raul García-Patrón. Limitations of optimization algorithms on noisy quantum devices. Nature Physics, 17 (11): 1221–1227, November 2021. ISSN 1745-2481. 10.1038/s41567-021-01356-3. URL https://www.nature.com/articles/s41567-021-01356-3. Number: 11 Publisher: Nature Publishing Group. https://doi.org/10.1038/s41567-021-01356-3 https://www.nature.com/articles/s41567-021-01356-3 [42] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li. Complete positivity order and relative entropy decay. Forum of Mathematics, Sigma, 13: e31, January 2025. ISSN 2050-5094. 10.1017/fms.2024.117. URL https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complete-positivity-order-and-relative-entropy-decay/7270B6BBA7891B612C00FAB066F1A380. https://doi.org/10.1017/fms.2024.117 https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complete-positivity-order-and-relative-entropy-decay/7270B6BBA7891B612C00FAB066F1A380 [43] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice. Communications in Mathematical Physics, 405 (2): 42, February 2024. ISSN 1432-0916. 10.1007/s00220-023-04869-5. URL https://doi.org/10.1007/s00220-023-04869-5. https://doi.org/10.1007/s00220-023-04869-5 [44] Chi-Fang Chen and Fernando G. S. L. Brandão. Fast Thermalization from the Eigenstate Thermalization Hypothesis. March 2023. 10.48550/arXiv.2112.07646. URL http://arxiv.org/abs/2112.07646. arXiv:2112.07646 [cond-mat, physics:math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.2112.07646 arXiv:2112.07646 [45] Nicholas LaRacuente. Quasi-factorization and multiplicative comparison of subalgebra-relative entropy. Journal of Mathematical Physics, 63 (12): 122203, December 2022. ISSN 0022-2488. 10.1063/5.0053698. URL https://doi.org/10.1063/5.0053698. https://doi.org/10.1063/5.0053698 [46] David W. Kribs and Robert W. Spekkens. Quantum error-correcting subsystems are unitarily recoverable subsystems. Physical Review A, 74 (4): 042329, October 2006. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.74.042329. URL https://doi.org/10.1103/PhysRevA.74.042329. https://doi.org/10.1103/PhysRevA.74.042329 [47] Man-Duen Choi, Nathaniel Johnston, and David W Kribs. The multiplicative domain in quantum error correction. Journal of Physics A: Mathematical and Theoretical, 42 (24): 245303, June 2009. ISSN 1751-8113, 1751-8121. 10.1088/1751-8113/42/24/245303. URL https://iopscience.iop.org/article/10.1088/1751-8113/42/24/245303. https://doi.org/10.1088/1751-8113/42/24/245303 [48] Nathaniel Johnston and David Kribs. Generalized multiplicative domains and quantum error correction. Proceedings of the American Mathematical Society, 139 (2): 627–639, February 2011. ISSN 0002-9939, 1088-6826. 10.1090/S0002-9939-2010-10556-7. URL https://www.ams.org/proc/2011-139-02/S0002-9939-2010-10556-7/. https://doi.org/10.1090/S0002-9939-2010-10556-7 https://www.ams.org/proc/2011-139-02/S0002-9939-2010-10556-7/ [49] Mizanur Rahaman. Multiplicative properties of quantum channels. Journal of Physics A: Mathematical and Theoretical, 50 (34): 345302, August 2017. ISSN 1751-8113, 1751-8121. 10.1088/1751-8121/aa7b57. URL https://iopscience.iop.org/article/10.1088/1751-8121/aa7b57. https://doi.org/10.1088/1751-8121/aa7b57 [50] D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-Free Subspaces for Quantum Computation.
Physical Review Letters, 81 (12): 2594–2597, September 1998. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.81.2594. URL https://doi.org/10.1103/PhysRevLett.81.2594. https://doi.org/10.1103/PhysRevLett.81.2594 [51] Julián Agredo, Franco Fagnola, and Rolando Rebolledo. Decoherence free subspaces of a quantum Markov semigroup. Journal of Mathematical Physics, 55 (11): 112201, November 2014. ISSN 0022-2488, 1089-7658. 10.1063/1.4901009. URL https://pubs.aip.org/jmp/article/55/11/112201/400172/Decoherence-free-subspaces-of-a-quantum-Markov. https://doi.org/10.1063/1.4901009 https://pubs.aip.org/jmp/article/55/11/112201/400172/Decoherence-free-subspaces-of-a-quantum-Markov [52] Daniel A. Lidar. Review of Decoherence-Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling.
In Quantum Information and Computation for Chemistry, pages 295–354. John Wiley & Sons, Ltd, 2014. ISBN 978-1-118-74263-1. 10.1002/9781118742631.ch11. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118742631.ch11. https://doi.org/10.1002/9781118742631.ch11 [53] Raffaella Carbone and Anna Jenčová. On Period, Cycles and Fixed Points of a Quantum Channel.
Annales Henri Poincaré, 21 (1): 155–188, January 2020. ISSN 1424-0661. 10.1007/s00023-019-00861-9. URL https://doi.org/10.1007/s00023-019-00861-9. https://doi.org/10.1007/s00023-019-00861-9 [54] John Watrous. The Theory of Quantum Information.
Cambridge University Press, Cambridge, 2018. ISBN 978-1-107-18056-7. 10.1017/9781316848142, online version at https://cs.uwaterloo.ca/%7Ewatrous/TQI/, accessed Jan 2023. URL https://www.cambridge.org/core/books/theory-of-quantum-information/AE4AA5638F808D2CFEB070C55431D897. https://www.cambridge.org/core/books/theory-of-quantum-information/AE4AA5638F808D2CFEB070C55431D897 [55] Marius Junge and Quanhua Xu. Noncommutative maximal ergodic theorems. Journal of the American Mathematical Society, 20 (2): 385–439, 2006. ISSN 0894-0347, 1088-6834. 10.1090/S0894-0347-06-00533-9. URL https://www.ams.org/jams/2007-20-02/S0894-0347-06-00533-9/. https://doi.org/10.1090/S0894-0347-06-00533-9 https://www.ams.org/jams/2007-20-02/S0894-0347-06-00533-9/ [56] B. Misra and E. C. G. Sudarshan. The Zeno’s paradox in quantum theory. Journal of Mathematical Physics, 18 (4): 756–763, 1977. ISSN 0022-2488. 10.1063/1.523304. URL https://doi.org/10.1063/1.523304. https://doi.org/10.1063/1.523304 [57] Norbert Barankai and Zoltán Zimborás. Generalized quantum Zeno dynamics and ergodic means. November 2018. 10.48550/arXiv.1811.02509. URL http://arxiv.org/abs/1811.02509. arXiv:1811.02509 [math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.1811.02509 arXiv:1811.02509 [58] Tim Möbus and Michael M. Wolf. Quantum Zeno effect generalized. Journal of Mathematical Physics, 60 (5): 052201, May 2019. ISSN 0022-2488. 10.1063/1.5090912. URL https://doi.org/10.1063/1.5090912. https://doi.org/10.1063/1.5090912 [59] Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato, Saverio Pascazio, and Kazuya Yuasa.
Quantum Zeno Dynamics from General Quantum Operations. Quantum, 4: 289, July 2020. 10.22331/q-2020-07-06-289. URL https://quantum-journal.org/papers/q-2020-07-06-289/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2020-07-06-289 https://quantum-journal.org/papers/q-2020-07-06-289/ [60] Simon Becker, Nilanjana Datta, and Robert Salzmann.
Quantum Zeno Effect in Open Quantum Systems.
Annales Henri Poincaré, 22 (11): 3795–3840, November 2021. ISSN 1424-0661. 10.1007/s00023-021-01075-8. URL https://doi.org/10.1007/s00023-021-01075-8. https://doi.org/10.1007/s00023-021-01075-8 [61] Tim Möbus and Cambyse Rouzé.
Optimal Convergence Rate in the Quantum Zeno Effect for Open Quantum Systems in Infinite Dimensions.
Annales Henri Poincaré, 24 (5): 1617–1659, May 2023. ISSN 1424-0661. 10.1007/s00023-022-01241-6. URL https://doi.org/10.1007/s00023-022-01241-6. https://doi.org/10.1007/s00023-022-01241-6 [62] Tosio Kato. On the Adiabatic Theorem of Quantum Mechanics. Journal of the Physical Society of Japan, 5 (6): 435–439, November 1950. ISSN 0031-9015. 10.1143/JPSJ.5.435. URL https://journals.jps.jp/doi/10.1143/JPSJ.5.435. Publisher: The Physical Society of Japan. https://doi.org/10.1143/JPSJ.5.435 [63] Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato, Saverio Pascazio, and Kazuya Yuasa.
Generalized Adiabatic Theorem and Strong-Coupling Limits. Quantum, 3: 152, June 2019. 10.22331/q-2019-06-12-152. URL https://quantum-journal.org/papers/q-2019-06-12-152/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2019-06-12-152 https://quantum-journal.org/papers/q-2019-06-12-152/ [64] Paolo Zanardi and Lorenzo Campos Venuti.
Coherent Quantum Dynamics in Steady-State Manifolds of Strongly Dissipative Systems.
Physical Review Letters, 113 (24): 240406, December 2014. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.113.240406. URL https://doi.org/10.1103/PhysRevLett.113.240406. https://doi.org/10.1103/PhysRevLett.113.240406 [65] Daniel Burgarth, Paolo Facchi, Giovanni Gramegna, and Kazuya Yuasa. One bound to rule them all: from Adiabatic to Zeno. Quantum, 6: 737, June 2022. 10.22331/q-2022-06-14-737. URL https://quantum-journal.org/papers/q-2022-06-14-737/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-06-14-737 https://quantum-journal.org/papers/q-2022-06-14-737/ [66] Masuo Suzuki. Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Communications in Mathematical Physics, 51 (2): 183–190, June 1976. ISSN 1432-0916. 10.1007/BF01609348. URL https://doi.org/10.1007/BF01609348. https://doi.org/10.1007/BF01609348 [67] Noam Erez, Yakir Aharonov, Benni Reznik, and Lev Vaidman. Correcting quantum errors with the Zeno effect. Physical Review A, 69 (6): 062315, June 2004. 10.1103/PhysRevA.69.062315. URL https://doi.org/10.1103/PhysRevA.69.062315. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.69.062315 [68] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Physical Review A, 58 (4): 2733–2744, October 1998. 10.1103/PhysRevA.58.2733. URL https://doi.org/10.1103/PhysRevA.58.2733. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.58.2733 [69] Lorenza Viola, Emanuel Knill, and Seth Lloyd. Dynamical Decoupling of Open Quantum Systems.
Physical Review Letters, 82 (12): 2417–2421, March 1999. 10.1103/PhysRevLett.82.2417. URL https://doi.org/10.1103/PhysRevLett.82.2417. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.82.2417 [70] John E. Gough and Hendra I. Nurdin. Can quantum Markov evolutions ever be dynamically decoupled? In 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pages 6155–6160, Melbourne, VIC, December 2017. IEEE. ISBN 9781509028733. 10.1109/CDC.2017.8264587. URL http://ieeexplore.ieee.org/document/8264587/. https://doi.org/10.1109/CDC.2017.8264587 http://ieeexplore.ieee.org/document/8264587/ [71] Almut Beige, Daniel Braun, Ben Tregenna, and Peter L. Knight.
Quantum Computing Using Dissipation to Remain in a Decoherence-Free Subspace.
Physical Review Letters, 85 (8): 1762–1765, August 2000. 10.1103/PhysRevLett.85.1762. URL https://doi.org/10.1103/PhysRevLett.85.1762. https://doi.org/10.1103/PhysRevLett.85.1762 [72] Yu-Jie Liu and Simon Lieu. Dissipative phase transitions and passive error correction. Physical Review A, 109 (2): 022422, February 2024. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.109.022422. URL https://doi.org/10.1103/PhysRevA.109.022422. https://doi.org/10.1103/PhysRevA.109.022422 [73] Solving the Lindblad dynamics of a qubit chain — Qiskit Dynamics 0.4.2 documentation, 2021. URL https://qiskit.org/ecosystem/dynamics/tutorials/Lindblad_dynamics_simulation.html. https://qiskit.org/ecosystem/dynamics/tutorials/Lindblad_dynamics_simulation.html [74] Clément L. Canonne. A short note on an inequality between KL and TV. August 2023. 10.48550/arXiv.2202.07198. URL http://arxiv.org/abs/2202.07198. arXiv:2202.07198 [math, stat] version: 2. https://doi.org/10.48550/arXiv.2202.07198 arXiv:2202.07198 [75] I. Vajda. Note on discrimination information and variation (Corresp.). IEEE Transactions on Information Theory, 16 (6): 771–773, November 1970. ISSN 1557-9654. 10.1109/TIT.1970.1054557. URL https://ieeexplore.ieee.org/document/1054557. https://doi.org/10.1109/TIT.1970.1054557 https://ieeexplore.ieee.org/document/1054557 [76] Nicholas Laracuente and Graeme Smith. Information Fragility or Robustness Under Quantum Channels. December 2023. 10.48550/arXiv.2312.17450. URL http://arxiv.org/abs/2312.17450. arXiv:2312.17450 [quant-ph]. https://doi.org/10.48550/arXiv.2312.17450 arXiv:2312.17450 [77] Prakash Murali, Jonathan M. Baker, Ali Javadi-Abhari, Frederic T. Chong, and Margaret Martonosi. Noise-Adaptive Compiler Mappings for Noisy Intermediate-Scale Quantum Computers. In Proceedings of the Twenty-Fourth International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS '19, pages 1015–1029, New York, NY, USA, April 2019. Association for Computing Machinery. ISBN 978-1-4503-6240-5. 10.1145/3297858.3304075. URL https://dl.acm.org/doi/10.1145/3297858.3304075. https://doi.org/10.1145/3297858.3304075 [78] Thomas Alexander, Naoki Kanazawa, Daniel J. Egger, Lauren Capelluto, Christopher J. Wood, Ali Javadi-Abhari, and David C. McKay. Qiskit pulse: programming quantum computers through the cloud with pulses. Quantum Science and Technology, 5 (4): 044006, August 2020. ISSN 2058-9565. 10.1088/2058-9565/aba404. URL https://dx.doi.org/10.1088/2058-9565/aba404. Publisher: IOP Publishing. https://doi.org/10.1088/2058-9565/aba404 [79] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver. A quantum engineer's guide to superconducting qubits.
Applied Physics Reviews, 6 (2): 021318, June 2019. ISSN 1931-9401. 10.1063/1.5089550. URL https://pubs.aip.org/apr/article/6/2/021318/570326/A-quantum-engineer-s-guide-to-superconducting. https://doi.org/10.1063/1.5089550 https://pubs.aip.org/apr/article/6/2/021318/570326/A-quantum-engineer-s-guide-to-superconducting [80] Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 945–957, New York, NY, USA, June 2023. Association for Computing Machinery. ISBN 978-1-4503-9913-5. 10.1145/3564246.3585234. URL https://dl.acm.org/doi/10.1145/3564246.3585234. https://doi.org/10.1145/3564246.3585234 [81] Stefan Walter, Andreas Nunnenkamp, and Christoph Bruder. Quantum Synchronization of a Driven Self-Sustained Oscillator.
Physical Review Letters, 112 (9): 094102, March 2014. 10.1103/PhysRevLett.112.094102. URL https://doi.org/10.1103/PhysRevLett.112.094102. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.112.094102 [82] Lior Ben Arosh, M. C. Cross, and Ron Lifshitz. Quantum limit cycles and the Rayleigh and van der Pol oscillators.
Physical Review Research, 3 (1): 013130, February 2021. 10.1103/PhysRevResearch.3.013130. URL https://doi.org/10.1103/PhysRevResearch.3.013130. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevResearch.3.013130Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-04 09:32:59: Could not fetch cited-by data for 10.22331/q-2026-03-04-2010 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-04 09:32:59: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractStates of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.Featured image: Relative entropy of one qubit within a two-qubit system to its completely mixed fixed point at a fixed time. The two qubits interact via the Hamiltonian as in Example 4.2 of the paper, but only the other qubit is exposed to dissipative noise. As the decay rate induced by that noise increases, the relative entropy shown dips, then rebounds.Popular summaryIn the absence of error correction or perfect isolation, a quantum system's exposure to its environment typically induces noise. It is known that when a system's dynamics arise entirely from noisy environmental interactions, its entropy approaches that of equilibrium at an exponential rate. However, many of these known results do not immediately apply when combining noise with internal processes. Herein, theories of exponential equilibration are extended to a wide class of such combined processes. When noise is restricted to a part of the system, it might still be spread by the system's internal dynamics. In such scenarios, extremely rapid noisy interactions actually reduce the system's decay rate toward equilibrium. This 'self-restriction' occurs because the noise process suppresses interactions between the highly noised part and the rest of the system.► BibTeX data@article{LaRacuente2026selfrestricting, doi = {10.22331/q-2026-03-04-2010}, url = {https://doi.org/10.22331/q-2026-03-04-2010}, title = {Self-restricting {N}oise and {E}xponential {R}elative {E}ntropy {D}ecay {U}nder {U}nital {Q}uantum {M}arkov {S}emigroups}, author = {LaRacuente, Nicholas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2010}, month = mar, year = {2026} }► References [1] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf. Introduction to quantum noise, measurement, and amplification. Reviews of Modern Physics, 82 (2): 1155–1208, April 2010. 10.1103/RevModPhys.82.1155. URL https://doi.org/10.1103/RevModPhys.82.1155. Publisher: American Physical Society. https://doi.org/10.1103/RevModPhys.82.1155 [2] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/q-2018-08-06-79. URL https://quantum-journal.org/papers/q-2018-08-06-79/. https://doi.org/10.22331/q-2018-08-06-79 https://quantum-journal.org/papers/q-2018-08-06-79/ [3] Mark M. Wilde.
Quantum Information Theory.
Cambridge University Press, Cambridge, 2 edition, 2017. ISBN 978-1-107-17616-4. 10.1017/9781316809976. URL https://www.cambridge.org/core/books/quantum-information-theory/247A740E156416531AA8CB97DFDAE438. https://doi.org/10.1017/9781316809976 https://www.cambridge.org/core/books/quantum-information-theory/247A740E156416531AA8CB97DFDAE438 [4] R Rebolledo. Decoherence of quantum Markov semigroups. Annales de l'Institut Henri Poincare (B) Probability and Statistics, 41 (3): 349–373, May 2005. ISSN 02460203. 10.1016/j.anihpb.2004.12.003. URL https://linkinghub.elsevier.com/retrieve/pii/S0246020305000257. https://doi.org/10.1016/j.anihpb.2004.12.003 https://linkinghub.elsevier.com/retrieve/pii/S0246020305000257 [5] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. Rapid Thermalization of Spin Chain Commuting Hamiltonians.
Physical Review Letters, 130 (6): 060401, February 2023. 10.1103/PhysRevLett.130.060401. URL https://doi.org/10.1103/PhysRevLett.130.060401. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.130.060401 [6] Andreas Bluhm, Ángela Capel, and Antonio Pérez-Hernández. Exponential decay of mutual information for Gibbs states of local Hamiltonians. Quantum, 6: 650, February 2022. 10.22331/q-2022-02-10-650. URL https://quantum-journal.org/papers/q-2022-02-10-650/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-02-10-650 https://quantum-journal.org/papers/q-2022-02-10-650/ [7] G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48 (2): 119–130, June 1976. ISSN 0010-3616, 1432-0916. 10.1007/BF01608499. URL http://link.springer.com/10.1007/BF01608499. https://doi.org/10.1007/BF01608499 [8] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. Journal of Mathematical Physics, 17 (5): 821–825, 1976. ISSN 0022-2488. 10.1063/1.522979. URL https://doi.org/10.1063/1.522979. https://doi.org/10.1063/1.522979 [9] Hisaharu Umegaki. Conditional expectation in an operator algebra. IV. Entropy and information.
Kodai Mathematical Seminar Reports, 14 (2): 59–85, 1962. ISSN 0023-2599. 10.2996/kmj/1138844604. URL https://projecteuclid.org/euclid.kmj/1138844604. https://doi.org/10.2996/kmj/1138844604 https://projecteuclid.org/euclid.kmj/1138844604 [10] Li Gao and Cambyse Rouzé.
Complete Entropic Inequalities for Quantum Markov Chains. Archive for Rational Mechanics and Analysis, 245 (1): 183–238, July 2022. ISSN 1432-0673. 10.1007/s00205-022-01785-1. URL https://doi.org/10.1007/s00205-022-01785-1. https://doi.org/10.1007/s00205-022-01785-1 [11] Li Gao, Marius Junge, and Haojian Li.
Geometric Approach Towards Complete Logarithmic Sobolev Inequalities. February 2021. 10.48550/arXiv.2102.04434. URL http://arxiv.org/abs/2102.04434. arXiv:2102.04434 [quant-ph]. https://doi.org/10.48550/arXiv.2102.04434 arXiv:2102.04434 [12] Marius Junge, Nicholas Laracuente, and Cambyse Rouzé. Stability of Logarithmic Sobolev Inequalities Under a Noncommutative Change of Measure. Journal of Statistical Physics, 190 (2): 30, December 2022. ISSN 1572-9613. 10.1007/s10955-022-03026-x. URL https://doi.org/10.1007/s10955-022-03026-x. https://doi.org/10.1007/s10955-022-03026-x [13] Li Gao, Marius Junge, and Nicholas LaRacuente. Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions.
Annales Henri Poincaré, 21 (11): 3409–3478, November 2020a. ISSN 1424-0661. 10.1007/s00023-020-00947-9. URL https://doi.org/10.1007/s00023-020-00947-9. https://doi.org/10.1007/s00023-020-00947-9 [14] Michael J. Kastoryano and Kristan Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. Journal of Mathematical Physics, 54 (5): 052202, May 2013. ISSN 0022-2488. 10.1063/1.4804995. URL https://doi.org/10.1063/1.4804995. https://doi.org/10.1063/1.4804995 [15] Ameur Dhahri, Franco Fagnola, and Rolando Rebolledo. The Decoherence-Free Subalgebra of a Quantum Markov Semigroup with Unbounded Generator.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 13 (03): 413–433, September 2010. ISSN 0219-0257, 1793-6306. 10.1142/S0219025710004176. URL https://www.worldscientific.com/doi/abs/10.1142/S0219025710004176. https://doi.org/10.1142/S0219025710004176 [16] Franco Fagnola, Emanuela Sasso, and Veronica Umanità. The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups. Journal of Mathematical Physics, 60 (7): 072703, July 2019. ISSN 0022-2488, 1089-7658. 10.1063/1.5030954. URL https://pubs.aip.org/jmp/article/60/7/072703/319785/The-role-of-the-atomic-decoherence-free-subalgebra. https://doi.org/10.1063/1.5030954 https://pubs.aip.org/jmp/article/60/7/072703/319785/The-role-of-the-atomic-decoherence-free-subalgebra [17] Vladislav Popkov, Simon Essink, Carlo Presilla, and Gunter Schütz. Effective quantum Zeno dynamics in dissipative quantum systems. Physical Review A, 98 (5): 052110, November 2018. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.98.052110. URL https://doi.org/10.1103/PhysRevA.98.052110. https://doi.org/10.1103/PhysRevA.98.052110 [18] Li Gao, Marius Junge, and Nicholas LaRacuente. Relative entropy for von Neumann subalgebras. International Journal of Mathematics, 31 (06): 2050046, June 2020b. ISSN 0129-167X. 10.1142/S0129167X20500469. URL https://www.worldscientific.com/doi/abs/10.1142/S0129167X20500469. Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0129167X20500469 [19] Mihai Pimsner and Sorin Popa. Entropy and index for subfactors. Annales scientifiques de l'École Normale Supérieure, 19 (1): 57–106, 1986. ISSN 1873-2151. 10.24033/asens.1504. URL http://www.numdam.org/item/?id=ASENS_1986_4_19_1_57_0. https://doi.org/10.24033/asens.1504 http://www.numdam.org/item/?id=ASENS_1986_4_19_1_57_0 [20] V. F. R. Jones. Index for subfactors. Inventiones mathematicae, 72 (1): 1–25, February 1983. ISSN 1432-1297. 10.1007/BF01389127. URL https://doi.org/10.1007/BF01389127. https://doi.org/10.1007/BF01389127 [21] Göran Lindblad. Completely positive maps and entropy inequalities. Communications in Mathematical Physics, 40 (2): 147–151, June 1975. ISSN 1432-0916. 10.1007/BF01609396. URL https://doi.org/10.1007/BF01609396. https://doi.org/10.1007/BF01609396 [22] Andreas Winter.
Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints. Communications in Mathematical Physics, 347 (1): 291–313, October 2016. ISSN 1432-0916. 10.1007/s00220-016-2609-8. URL https://doi.org/10.1007/s00220-016-2609-8. https://doi.org/10.1007/s00220-016-2609-8 [23] Andreas Bluhm, Ángela Capel, Paul Gondolf, and Antonio Pérez-Hernández. Continuity of Quantum Entropic Quantities via Almost Convexity. IEEE Transactions on Information Theory, 69 (9): 5869–5901, September 2023. ISSN 0018-9448, 1557-9654. 10.1109/TIT.2023.3277892. URL https://ieeexplore.ieee.org/document/10129917/. https://doi.org/10.1109/TIT.2023.3277892 https://ieeexplore.ieee.org/document/10129917/ [24] Andreas Bluhm, Ángela Capel, Paul Gondolf, and Antonio Pérez-Hernández. Corrections to “Continuity of Quantum Entropic Quantities via Almost Convexity”. IEEE Transactions on Information Theory, 70 (7): 5410–5411, July 2024. ISSN 0018-9448, 1557-9654. 10.1109/TIT.2024.3390839. URL https://ieeexplore.ieee.org/document/10504886/. https://doi.org/10.1109/TIT.2024.3390839 https://ieeexplore.ieee.org/document/10504886/ [25] J. Lin. Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37 (1): 145–151, January 1991. ISSN 00189448. 10.1109/18.61115. URL http://ieeexplore.ieee.org/document/61115/. https://doi.org/10.1109/18.61115 http://ieeexplore.ieee.org/document/61115/ [26] M. B. Hastings. Superadditivity of communication capacity using entangled inputs. Nature Physics, 5 (4): 255–257, April 2009. ISSN 1745-2481. 10.1038/nphys1224. URL https://www.nature.com/articles/nphys1224. Number: 4 Publisher: Nature Publishing Group. https://doi.org/10.1038/nphys1224 https://www.nature.com/articles/nphys1224 [27] Dénes Petz. Sufficiency of Channels Over Von Neumann Algebras.
The Quarterly Journal of Mathematics, 39 (1): 97–108, March 1988. ISSN 0033-5606. 10.1093/qmath/39.1.97. URL https://doi.org/10.1093/qmath/39.1.97. https://doi.org/10.1093/qmath/39.1.97 [28] Maximilian Schlosshauer. Quantum decoherence. Physics Reports, 831: 1–57, October 2019. ISSN 0370-1573. 10.1016/j.physrep.2019.10.001. URL https://www.sciencedirect.com/science/article/pii/S0370157319303084. https://doi.org/10.1016/j.physrep.2019.10.001 https://www.sciencedirect.com/science/article/pii/S0370157319303084 [29] Dieter Suter and Gonzalo A. Álvarez. Colloquium: Protecting quantum information against environmental noise. Reviews of Modern Physics, 88 (4): 041001, October 2016. 10.1103/RevModPhys.88.041001. URL https://doi.org/10.1103/RevModPhys.88.041001. Publisher: American Physical Society. https://doi.org/10.1103/RevModPhys.88.041001 [30] M. Beau, J. Kiukas, I. L. Egusquiza, and A. del Campo.
Nonexponential Quantum Decay under Environmental Decoherence.
Physical Review Letters, 119 (13): 130401, September 2017. 10.1103/PhysRevLett.119.130401. URL https://doi.org/10.1103/PhysRevLett.119.130401. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.119.130401 [31] I.A. Aloisio, G.A.L. White, C.D. Hill, and K. Modi. Sampling Complexity of Open Quantum Systems. PRX Quantum, 4 (2): 020310, April 2023. 10.1103/PRXQuantum.4.020310. URL https://doi.org/10.1103/PRXQuantum.4.020310. Publisher: American Physical Society. https://doi.org/10.1103/PRXQuantum.4.020310 [32] Patrick Hayden and Jonathan Sorce. A canonical Hamiltonian for open quantum systems. Journal of Physics A: Mathematical and Theoretical, 55 (22): 225302, May 2022. ISSN 1751-8121. 10.1088/1751-8121/ac65c2. URL https://dx.doi.org/10.1088/1751-8121/ac65c2. Publisher: IOP Publishing. https://doi.org/10.1088/1751-8121/ac65c2 [33] Marco Merkli. Dynamics of Open Quantum Systems II, Markovian Approximation. Quantum, 6: 616, January 2022. 10.22331/q-2022-01-03-616. URL https://quantum-journal.org/papers/q-2022-01-03-616/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-01-03-616 https://quantum-journal.org/papers/q-2022-01-03-616/ [34] Anton Arnold, Peter Markowich, Giuseppe Toscani, and Andreas Unterreiter. On logarithmic Sobolev inequalities, Csiszar-Kullback inequalities, and the rate of convergence to equilibrium for Fokker-Planck type equations. Preprint-Reihe Mathematik, 592: 77, 1998. [35] Sergey Bobkov and Prasad Tetali. Modified log-sobolev inequalities, mixing and hypercontractivity. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, STOC '03, pages 287–296, New York, NY, USA, June 2003. Association for Computing Machinery. ISBN 978-1-58113-674-6. 10.1145/780542.780586. URL https://dl.acm.org/doi/10.1145/780542.780586. https://doi.org/10.1145/780542.780586 [36] Leonard Gross.
Logarithmic Sobolev Inequalities. American Journal of Mathematics, 97 (4): 1061–1083, 1975a. ISSN 0002-9327. 10.2307/2373688. URL https://www.jstor.org/stable/2373688. Publisher: Johns Hopkins University Press. https://doi.org/10.2307/2373688 https://www.jstor.org/stable/2373688 [37] Leonard Gross. Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form.
Duke Mathematical Journal, 42 (3): 383–396, September 1975b. ISSN 0012-7094, 1547-7398. 10.1215/S0012-7094-75-04237-4. URL https://projecteuclid.org/journals/duke-mathematical-journal/volume-42/issue-3/Hypercontractivity-and-logarithmic-Sobolev-inequalities-for-the-Clifford-Dirichlet-form/10.1215/S0012-7094-75-04237-4.full. https://doi.org/10.1215/S0012-7094-75-04237-4 [38] Ivan Bardet and Cambyse Rouzé. Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates.
Annales Henri Poincaré, 23 (11): 3839–3903, November 2022. ISSN 1424-0661. 10.1007/s00023-022-01196-8. URL https://doi.org/10.1007/s00023-022-01196-8. https://doi.org/10.1007/s00023-022-01196-8 [39] Ivan Bardet. Estimating the decoherence time using non-commutative Functional Inequalities. October 2017. 10.48550/arXiv.1710.01039. URL http://arxiv.org/abs/1710.01039. arXiv:1710.01039 [math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.1710.01039 arXiv:1710.01039 [40] Ivan Bardet, Marius Junge, Nicholas Laracuente, Cambyse Rouzé, and Daniel Stilck França.
Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups. IEEE Transactions on Information Theory, 67 (5): 2878–2909, May 2021. ISSN 1557-9654. 10.1109/TIT.2021.3065452. Conference Name: IEEE Transactions on Information Theory. https://doi.org/10.1109/TIT.2021.3065452 [41] Daniel Stilck França and Raul García-Patrón. Limitations of optimization algorithms on noisy quantum devices. Nature Physics, 17 (11): 1221–1227, November 2021. ISSN 1745-2481. 10.1038/s41567-021-01356-3. URL https://www.nature.com/articles/s41567-021-01356-3. Number: 11 Publisher: Nature Publishing Group. https://doi.org/10.1038/s41567-021-01356-3 https://www.nature.com/articles/s41567-021-01356-3 [42] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li. Complete positivity order and relative entropy decay. Forum of Mathematics, Sigma, 13: e31, January 2025. ISSN 2050-5094. 10.1017/fms.2024.117. URL https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complete-positivity-order-and-relative-entropy-decay/7270B6BBA7891B612C00FAB066F1A380. https://doi.org/10.1017/fms.2024.117 https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/complete-positivity-order-and-relative-entropy-decay/7270B6BBA7891B612C00FAB066F1A380 [43] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice. Communications in Mathematical Physics, 405 (2): 42, February 2024. ISSN 1432-0916. 10.1007/s00220-023-04869-5. URL https://doi.org/10.1007/s00220-023-04869-5. https://doi.org/10.1007/s00220-023-04869-5 [44] Chi-Fang Chen and Fernando G. S. L. Brandão. Fast Thermalization from the Eigenstate Thermalization Hypothesis. March 2023. 10.48550/arXiv.2112.07646. URL http://arxiv.org/abs/2112.07646. arXiv:2112.07646 [cond-mat, physics:math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.2112.07646 arXiv:2112.07646 [45] Nicholas LaRacuente. Quasi-factorization and multiplicative comparison of subalgebra-relative entropy. Journal of Mathematical Physics, 63 (12): 122203, December 2022. ISSN 0022-2488. 10.1063/5.0053698. URL https://doi.org/10.1063/5.0053698. https://doi.org/10.1063/5.0053698 [46] David W. Kribs and Robert W. Spekkens. Quantum error-correcting subsystems are unitarily recoverable subsystems. Physical Review A, 74 (4): 042329, October 2006. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.74.042329. URL https://doi.org/10.1103/PhysRevA.74.042329. https://doi.org/10.1103/PhysRevA.74.042329 [47] Man-Duen Choi, Nathaniel Johnston, and David W Kribs. The multiplicative domain in quantum error correction. Journal of Physics A: Mathematical and Theoretical, 42 (24): 245303, June 2009. ISSN 1751-8113, 1751-8121. 10.1088/1751-8113/42/24/245303. URL https://iopscience.iop.org/article/10.1088/1751-8113/42/24/245303. https://doi.org/10.1088/1751-8113/42/24/245303 [48] Nathaniel Johnston and David Kribs. Generalized multiplicative domains and quantum error correction. Proceedings of the American Mathematical Society, 139 (2): 627–639, February 2011. ISSN 0002-9939, 1088-6826. 10.1090/S0002-9939-2010-10556-7. URL https://www.ams.org/proc/2011-139-02/S0002-9939-2010-10556-7/. https://doi.org/10.1090/S0002-9939-2010-10556-7 https://www.ams.org/proc/2011-139-02/S0002-9939-2010-10556-7/ [49] Mizanur Rahaman. Multiplicative properties of quantum channels. Journal of Physics A: Mathematical and Theoretical, 50 (34): 345302, August 2017. ISSN 1751-8113, 1751-8121. 10.1088/1751-8121/aa7b57. URL https://iopscience.iop.org/article/10.1088/1751-8121/aa7b57. https://doi.org/10.1088/1751-8121/aa7b57 [50] D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-Free Subspaces for Quantum Computation.
Physical Review Letters, 81 (12): 2594–2597, September 1998. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.81.2594. URL https://doi.org/10.1103/PhysRevLett.81.2594. https://doi.org/10.1103/PhysRevLett.81.2594 [51] Julián Agredo, Franco Fagnola, and Rolando Rebolledo. Decoherence free subspaces of a quantum Markov semigroup. Journal of Mathematical Physics, 55 (11): 112201, November 2014. ISSN 0022-2488, 1089-7658. 10.1063/1.4901009. URL https://pubs.aip.org/jmp/article/55/11/112201/400172/Decoherence-free-subspaces-of-a-quantum-Markov. https://doi.org/10.1063/1.4901009 https://pubs.aip.org/jmp/article/55/11/112201/400172/Decoherence-free-subspaces-of-a-quantum-Markov [52] Daniel A. Lidar. Review of Decoherence-Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling.
In Quantum Information and Computation for Chemistry, pages 295–354. John Wiley & Sons, Ltd, 2014. ISBN 978-1-118-74263-1. 10.1002/9781118742631.ch11. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118742631.ch11. https://doi.org/10.1002/9781118742631.ch11 [53] Raffaella Carbone and Anna Jenčová. On Period, Cycles and Fixed Points of a Quantum Channel.
Annales Henri Poincaré, 21 (1): 155–188, January 2020. ISSN 1424-0661. 10.1007/s00023-019-00861-9. URL https://doi.org/10.1007/s00023-019-00861-9. https://doi.org/10.1007/s00023-019-00861-9 [54] John Watrous. The Theory of Quantum Information.
Cambridge University Press, Cambridge, 2018. ISBN 978-1-107-18056-7. 10.1017/9781316848142, online version at https://cs.uwaterloo.ca/%7Ewatrous/TQI/, accessed Jan 2023. URL https://www.cambridge.org/core/books/theory-of-quantum-information/AE4AA5638F808D2CFEB070C55431D897. https://www.cambridge.org/core/books/theory-of-quantum-information/AE4AA5638F808D2CFEB070C55431D897 [55] Marius Junge and Quanhua Xu. Noncommutative maximal ergodic theorems. Journal of the American Mathematical Society, 20 (2): 385–439, 2006. ISSN 0894-0347, 1088-6834. 10.1090/S0894-0347-06-00533-9. URL https://www.ams.org/jams/2007-20-02/S0894-0347-06-00533-9/. https://doi.org/10.1090/S0894-0347-06-00533-9 https://www.ams.org/jams/2007-20-02/S0894-0347-06-00533-9/ [56] B. Misra and E. C. G. Sudarshan. The Zeno’s paradox in quantum theory. Journal of Mathematical Physics, 18 (4): 756–763, 1977. ISSN 0022-2488. 10.1063/1.523304. URL https://doi.org/10.1063/1.523304. https://doi.org/10.1063/1.523304 [57] Norbert Barankai and Zoltán Zimborás. Generalized quantum Zeno dynamics and ergodic means. November 2018. 10.48550/arXiv.1811.02509. URL http://arxiv.org/abs/1811.02509. arXiv:1811.02509 [math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.1811.02509 arXiv:1811.02509 [58] Tim Möbus and Michael M. Wolf. Quantum Zeno effect generalized. Journal of Mathematical Physics, 60 (5): 052201, May 2019. ISSN 0022-2488. 10.1063/1.5090912. URL https://doi.org/10.1063/1.5090912. https://doi.org/10.1063/1.5090912 [59] Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato, Saverio Pascazio, and Kazuya Yuasa.
Quantum Zeno Dynamics from General Quantum Operations. Quantum, 4: 289, July 2020. 10.22331/q-2020-07-06-289. URL https://quantum-journal.org/papers/q-2020-07-06-289/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2020-07-06-289 https://quantum-journal.org/papers/q-2020-07-06-289/ [60] Simon Becker, Nilanjana Datta, and Robert Salzmann.
Quantum Zeno Effect in Open Quantum Systems.
Annales Henri Poincaré, 22 (11): 3795–3840, November 2021. ISSN 1424-0661. 10.1007/s00023-021-01075-8. URL https://doi.org/10.1007/s00023-021-01075-8. https://doi.org/10.1007/s00023-021-01075-8 [61] Tim Möbus and Cambyse Rouzé.
Optimal Convergence Rate in the Quantum Zeno Effect for Open Quantum Systems in Infinite Dimensions.
Annales Henri Poincaré, 24 (5): 1617–1659, May 2023. ISSN 1424-0661. 10.1007/s00023-022-01241-6. URL https://doi.org/10.1007/s00023-022-01241-6. https://doi.org/10.1007/s00023-022-01241-6 [62] Tosio Kato. On the Adiabatic Theorem of Quantum Mechanics. Journal of the Physical Society of Japan, 5 (6): 435–439, November 1950. ISSN 0031-9015. 10.1143/JPSJ.5.435. URL https://journals.jps.jp/doi/10.1143/JPSJ.5.435. Publisher: The Physical Society of Japan. https://doi.org/10.1143/JPSJ.5.435 [63] Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato, Saverio Pascazio, and Kazuya Yuasa.
Generalized Adiabatic Theorem and Strong-Coupling Limits. Quantum, 3: 152, June 2019. 10.22331/q-2019-06-12-152. URL https://quantum-journal.org/papers/q-2019-06-12-152/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2019-06-12-152 https://quantum-journal.org/papers/q-2019-06-12-152/ [64] Paolo Zanardi and Lorenzo Campos Venuti.
Coherent Quantum Dynamics in Steady-State Manifolds of Strongly Dissipative Systems.
Physical Review Letters, 113 (24): 240406, December 2014. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.113.240406. URL https://doi.org/10.1103/PhysRevLett.113.240406. https://doi.org/10.1103/PhysRevLett.113.240406 [65] Daniel Burgarth, Paolo Facchi, Giovanni Gramegna, and Kazuya Yuasa. One bound to rule them all: from Adiabatic to Zeno. Quantum, 6: 737, June 2022. 10.22331/q-2022-06-14-737. URL https://quantum-journal.org/papers/q-2022-06-14-737/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. https://doi.org/10.22331/q-2022-06-14-737 https://quantum-journal.org/papers/q-2022-06-14-737/ [66] Masuo Suzuki. Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Communications in Mathematical Physics, 51 (2): 183–190, June 1976. ISSN 1432-0916. 10.1007/BF01609348. URL https://doi.org/10.1007/BF01609348. https://doi.org/10.1007/BF01609348 [67] Noam Erez, Yakir Aharonov, Benni Reznik, and Lev Vaidman. Correcting quantum errors with the Zeno effect. Physical Review A, 69 (6): 062315, June 2004. 10.1103/PhysRevA.69.062315. URL https://doi.org/10.1103/PhysRevA.69.062315. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.69.062315 [68] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Physical Review A, 58 (4): 2733–2744, October 1998. 10.1103/PhysRevA.58.2733. URL https://doi.org/10.1103/PhysRevA.58.2733. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.58.2733 [69] Lorenza Viola, Emanuel Knill, and Seth Lloyd. Dynamical Decoupling of Open Quantum Systems.
Physical Review Letters, 82 (12): 2417–2421, March 1999. 10.1103/PhysRevLett.82.2417. URL https://doi.org/10.1103/PhysRevLett.82.2417. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.82.2417 [70] John E. Gough and Hendra I. Nurdin. Can quantum Markov evolutions ever be dynamically decoupled? In 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pages 6155–6160, Melbourne, VIC, December 2017. IEEE. ISBN 9781509028733. 10.1109/CDC.2017.8264587. URL http://ieeexplore.ieee.org/document/8264587/. https://doi.org/10.1109/CDC.2017.8264587 http://ieeexplore.ieee.org/document/8264587/ [71] Almut Beige, Daniel Braun, Ben Tregenna, and Peter L. Knight.
Quantum Computing Using Dissipation to Remain in a Decoherence-Free Subspace.
Physical Review Letters, 85 (8): 1762–1765, August 2000. 10.1103/PhysRevLett.85.1762. URL https://doi.org/10.1103/PhysRevLett.85.1762. https://doi.org/10.1103/PhysRevLett.85.1762 [72] Yu-Jie Liu and Simon Lieu. Dissipative phase transitions and passive error correction. Physical Review A, 109 (2): 022422, February 2024. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.109.022422. URL https://doi.org/10.1103/PhysRevA.109.022422. https://doi.org/10.1103/PhysRevA.109.022422 [73] Solving the Lindblad dynamics of a qubit chain — Qiskit Dynamics 0.4.2 documentation, 2021. URL https://qiskit.org/ecosystem/dynamics/tutorials/Lindblad_dynamics_simulation.html. https://qiskit.org/ecosystem/dynamics/tutorials/Lindblad_dynamics_simulation.html [74] Clément L. Canonne. A short note on an inequality between KL and TV. August 2023. 10.48550/arXiv.2202.07198. URL http://arxiv.org/abs/2202.07198. arXiv:2202.07198 [math, stat] version: 2. https://doi.org/10.48550/arXiv.2202.07198 arXiv:2202.07198 [75] I. Vajda. Note on discrimination information and variation (Corresp.). IEEE Transactions on Information Theory, 16 (6): 771–773, November 1970. ISSN 1557-9654. 10.1109/TIT.1970.1054557. URL https://ieeexplore.ieee.org/document/1054557. https://doi.org/10.1109/TIT.1970.1054557 https://ieeexplore.ieee.org/document/1054557 [76] Nicholas Laracuente and Graeme Smith. Information Fragility or Robustness Under Quantum Channels. December 2023. 10.48550/arXiv.2312.17450. URL http://arxiv.org/abs/2312.17450. arXiv:2312.17450 [quant-ph]. https://doi.org/10.48550/arXiv.2312.17450 arXiv:2312.17450 [77] Prakash Murali, Jonathan M. Baker, Ali Javadi-Abhari, Frederic T. Chong, and Margaret Martonosi. Noise-Adaptive Compiler Mappings for Noisy Intermediate-Scale Quantum Computers. In Proceedings of the Twenty-Fourth International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS '19, pages 1015–1029, New York, NY, USA, April 2019. Association for Computing Machinery. ISBN 978-1-4503-6240-5. 10.1145/3297858.3304075. URL https://dl.acm.org/doi/10.1145/3297858.3304075. https://doi.org/10.1145/3297858.3304075 [78] Thomas Alexander, Naoki Kanazawa, Daniel J. Egger, Lauren Capelluto, Christopher J. Wood, Ali Javadi-Abhari, and David C. McKay. Qiskit pulse: programming quantum computers through the cloud with pulses. Quantum Science and Technology, 5 (4): 044006, August 2020. ISSN 2058-9565. 10.1088/2058-9565/aba404. URL https://dx.doi.org/10.1088/2058-9565/aba404. Publisher: IOP Publishing. https://doi.org/10.1088/2058-9565/aba404 [79] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver. A quantum engineer's guide to superconducting qubits.
Applied Physics Reviews, 6 (2): 021318, June 2019. ISSN 1931-9401. 10.1063/1.5089550. URL https://pubs.aip.org/apr/article/6/2/021318/570326/A-quantum-engineer-s-guide-to-superconducting. https://doi.org/10.1063/1.5089550 https://pubs.aip.org/apr/article/6/2/021318/570326/A-quantum-engineer-s-guide-to-superconducting [80] Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 945–957, New York, NY, USA, June 2023. Association for Computing Machinery. ISBN 978-1-4503-9913-5. 10.1145/3564246.3585234. URL https://dl.acm.org/doi/10.1145/3564246.3585234. https://doi.org/10.1145/3564246.3585234 [81] Stefan Walter, Andreas Nunnenkamp, and Christoph Bruder. Quantum Synchronization of a Driven Self-Sustained Oscillator.
Physical Review Letters, 112 (9): 094102, March 2014. 10.1103/PhysRevLett.112.094102. URL https://doi.org/10.1103/PhysRevLett.112.094102. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.112.094102 [82] Lior Ben Arosh, M. C. Cross, and Ron Lifshitz. Quantum limit cycles and the Rayleigh and van der Pol oscillators.
Physical Review Research, 3 (1): 013130, February 2021. 10.1103/PhysRevResearch.3.013130. URL https://doi.org/10.1103/PhysRevResearch.3.013130. Publisher: American Physical Society. https://doi.org/10.1103/PhysRevResearch.3.013130Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-04 09:32:59: Could not fetch cited-by data for 10.22331/q-2026-03-04-2010 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-04 09:32:59: No response from ADS or unable to decode the received json data when getting the list of citing works.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.