Nonlinear extension of the quantum dynamical semigroup

Jakub Rembieliński; Paweł Caban

doi:10.22331/q-2021-03-23-420

Nonlinear extension of the quantum dynamical semigroup

Jakub Rembieliński and Paweł Caban

Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149/153, 90-236 Łódź, Poland

Published:	2021-03-23, volume 5, page 420
Eprint:	arXiv:2003.09170v3
Doi:	https://doi.org/10.22331/q-2021-03-23-420
Citation:	Quantum 5, 420 (2021).

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Abstract

In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kossakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the general qubit evolution in our model as well as an extension of the Jaynes-Cummings model. We apply our formalism to spin density matrix of a charged particle moving in the electromagnetic field as well as to flavor evolution of solar neutrinos.

Featured image: Evolution of a relativistic qubit in the electromagnetic field

► BibTeX data

@article{Rembielinski2021nonlinearextension,
  doi = {10.22331/q-2021-03-23-420},
  url = {https://doi.org/10.22331/q-2021-03-23-420},
  title = {Nonlinear extension of the quantum dynamical semigroup},
  author = {Rembieli{\'{n}}ski, Jakub and Caban, Pawe{\l{}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {420},
  month = mar,
  year = {2021}
}

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[1] Jakub Rembieliński and Jacek Ciborowski, "On nonlinear description of neutrino flavour evolution in solar matter", Annals of Physics 458, 169481 (2023).

[2] Efstratios Koukoutsis, Panagiotis Papagiannis, Kyriakos Hizanidis, Abhay K. Ram, George Vahala, óscar Amaro, Llucas I Iñigo Gamiz, and Dimosthenis Vallis, "Quantum Implementation of Non-unitary Operations with Biorthogonal Representations", Quantum Information & Computation 25 2, 141 (2025).

[3] Efstratios Koukoutsis, Kyriakos Hizanidis, Abhay K. Ram, and George Vahala, "Quantum simulation of dissipation for Maxwell equations in dispersive media", Future Generation Computer Systems 159, 221 (2024).

[4] Yu Cao and Jianfeng Lu, "Structure-Preserving Numerical Schemes for Lindblad Equations", Journal of Scientific Computing 102 1, 27 (2025).

[5] Eyal Buks, "Disentanglement and a nonlinear Schrödinger equation", Journal of Physics A: Mathematical and Theoretical 55 35, 355303 (2022).

[6] Michael R. Geller, "Proposal for a Lorenz qubit", Scientific Reports 13 1, 14106 (2023).

[7] Marta Emilia Bielińska, Michał Eckstein, and Paweł Horodecki, "Superluminal signalling and chaos in nonlinear quantum dynamics", New Journal of Physics 27 5, 053002 (2025).

[8] Michael R. Geller, "Fast Quantum State Discrimination with Nonlinear Positive Trace‐Preserving Channels", Advanced Quantum Technologies 6 6, 2200156 (2023).

[9] Maxwell Aifer, Juzar Thingna, and Sebastian Deffner, "Energetic Cost for Speedy Synchronization in Non-Hermitian Quantum Dynamics", Physical Review Letters 133 2, 020401 (2024).

[10] Maria Danelli and Matteo G. A. Paris, "Are intrinsic decoherence models physical theories?", Europhysics Letters 149 5, 50001 (2025).

[11] Xu Zhang and Guanrong Chen, "From classical chaotic systems to quantum qubits and qutrits", Chaos: An Interdisciplinary Journal of Nonlinear Science 35 11, 111102 (2025).

[12] Eyal Buks, "Spontaneous Disentanglement and Thermalization", Advanced Quantum Technologies 7 5, 2400036 (2024).

[13] Eyal Buks, "Spontaneous Collapse by Entanglement Suppression", Advanced Quantum Technologies 6 9, 2300103 (2023).

[14] Pablo Martinez-Azcona, Aritra Kundu, Avadh Saxena, Adolfo del Campo, and Aurélia Chenu, "Quantum Dynamics with Stochastic Non-Hermitian Hamiltonians", Physical Review Letters 135 1, 010402 (2025).

[15] Eyal Buks, "Partial Disentanglement in a Multipartite System", Advanced Quantum Technologies 6 12, 2300170 (2023).

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[17] Rohit Kishan Ray and Gian Paolo Beretta, "No-Signaling in Steepest Entropy Ascent: A Nonlinear, Non-Local, Non-Equilibrium Quantum Dynamics of Composite Systems Strongly Compatible with the Second Law", Entropy 27 10, 1018 (2025).

[18] Krzysztof Kowalski, "Linear and integrable nonlinear evolution of the qutrit", Quantum Information Processing 19 5, 145 (2020).

[19] Efstratios Koukoutsis, Kyriakos Hizanidis, Abhay K. Ram, and George Vahala, "Quantum simulation of dissipation for Maxwell equations in dispersive media", arXiv:2308.00056, (2023).

[20] Jakub Rembieliński and Karol Ławniczak, "Quantum selective measurement as a quasilinear evolution", arXiv:2605.13739, (2026).

[21] Jakub Rembieliński and Karol Ławniczak, "Quasilinear evolution versus von Neumann selective measurement", arXiv:2605.13756, (2026).

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