Abstract
In the present work we review and refine some results about fixed points of semigroups of quantum channels. Noncommutative potential theory enables us to show that the set of fixed points of a recurrent semigroup is a \(W^*\)-algebra; aside from the intrinsic interest of this result, it brings an improvement in the study of fixed points by means of absorption operators (a noncommutative generalization of absorption probabilities): under the assumption of absorbing recurrent space (hence allowing non-trivial transient space) we can provide a description of the fixed points set and a probabilistic characterization of when it is a \(W^*\)-algebra in terms of absorption operators. Moreover we are able to exhibit an example of a recurrent semigroup which does not admit a decomposition of the Hilbert space into orthogonal minimal invariant domains (contrarily to the case of classical Markov chains and positive recurrent semigroups of quantum channels).
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References
Albert, V.: Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states. Quantum 3, 151 (2019). https://doi.org/10.22331/q-2019-06-06-151
Arias, A., Gheondea, A., Gudder, S.: Fixed points of quantum operations. J. Math. Phys. 43, 5872–5881 (2002). https://doi.org/10.1063/1.1519669
Carbone, R., Girotti, F.: Absorption in invariant domains for semigroups of quantum channels. Ann. Henri Poincaré 22, 2497–2530 (2021). https://doi.org/10.1007/s00023-021-01016-5
Carbone, R., Jenčová, A.: Cycles and fixed points of a quantum channel. Ann. Henri Poincaré 21, 155–188 (2020). https://doi.org/10.1007/s00023-019-00861-9
Carbone, R., Pautrat, Y.: Open quantum random walks: reducibility, period, ergodic properties. Ann. Henri Poincaré 17, 99–135 (2016). https://doi.org/10.1007/s00023-015-0396-y
Carbone, R., Pautrat, Y.: Irreducible decompositions and stationary states of quantum channels. Rep. Math. Phys. 77, 293–313 (2016). https://doi.org/10.1016/S0034-4877(16)30032-5
Davies, E.B.: Quantum stochastic processes. II Comm. Math. Phys. 19, 83–105 (1970)
Fagnola, F., Rebolledo, R.: Transience and recurrence of quantum Markov semigroups. Probab. Theory Relat. Fields 126, 289–306 (2003). https://doi.org/10.1007/s00440-003-0268-0
Fagnola, F., Sasso, E., Umanità, V.: The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups. J. Math. Phys. 60, 072703 (2019). https://doi.org/10.1063/1.5030954
Frigerio, A., Verri, M.: Long-time asymptotic properties of dynamical semigroups on \(W^{\ast }\)-algebras. Math. Z. 180, 275–286 (1982). https://doi.org/10.1007/BF01318911
Gärtner, A., Kümmerer, B.: A coherent approach to recurrence and transience for quantum Markov operators. arXiv: 1211.6876 (2012) http://adsabs.harvard.edu/abs/2012arXiv1211.6876G
Gheondea, A.: Symmetries versus conservation laws in dynamical quantum systems: a unifying approach through propagation of fixed points. Ann. Henri Poincaré 19, 1787–1816 (2018). https://doi.org/10.1007/s00023-018-0666-6
Jenčová, A., Petz, D.: Sufficiency in quantum statistical inference. Commun. Math. Phys. 263, 259–276 (2006). https://doi.org/10.1007/s00220-005-1510-7
Emanuela Sasso, E., Umanita, V.: The general structure of the Decoherence-free subalgebra for uniformly continuous Quantum Markov semigroups. arXiv: 2101.05121 (2021)
Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (2002)
Tomiyama, J.: On the projection of norm one in \(W^*\)-algebras. III. Tohoku Math. J. 11, 125–129 (1959). https://doi.org/10.2748/tmj/1178244633
Umanità, V.: Classification and decomposition of quantum Markov semigroups. Probab. Theory Relat. Fields 134, 603–623 (2006). https://doi.org/10.1007/s00440-005-0450-7
Acknowledgements
The author acknowledges the support of the INDAM GNAMPA project 2020 “Evoluzioni markoviane quantistiche” and of the Italian Ministry of Education, University and Research (MIUR) for the Dipartimenti di Eccellenza Program (2018- 2022)-Dept. of Mathematics “F. Casorati”, University of Pavia. Special acknowledgments go to the organizers of the \(41^\text {st}\) International Conference on Quantum Probability and Related Topics and to Raffaella Carbone for introducing the author to the problem and for many useful suggestions.
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Girotti, F. (2022). Absorption and Fixed Points for Semigroups of Quantum Channels. In: Accardi, L., Mukhamedov, F., Al Rawashdeh, A. (eds) Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021. Springer Proceedings in Mathematics & Statistics, vol 390. Springer, Cham. https://doi.org/10.1007/978-3-031-06170-7_10
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