# Symmetries and monotones in Markovian quantum dynamics

Georgios Styliaris and Paolo Zanardi

Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA

### Abstract

What can one infer about the dynamical evolution of quantum systems just by symmetry considerations? For Markovian dynamics in finite dimensions, we present a simple construction that assigns to each symmetry of the generator a family of scalar functions over quantum states that are monotonic under the time evolution. The aforementioned monotones can be utilized to identify states that are non-reachable from an initial state by the time evolution and include all constraints imposed by conserved quantities, providing a generalization of Noether's theorem for this class of dynamics. As a special case, the generator itself can be considered a symmetry, resulting in non-trivial constraints over the time evolution, even if all conserved quantities trivialize. The construction utilizes tools from quantum information-geometry, mainly the theory of monotone Riemannian metrics. We analyze the prototypical cases of dephasing and Davies generators.

### ► References

[1] Lev D. Landau and Evgeny M. Lifshitz. Classical Mechanics. Addison-Wesley, 1959.

[2] Jun John Sakurai and Jim Napolitano. Modern quantum mechanics. Cambridge University Press, 2017. doi:10.1017/​9781108499996.
https:/​/​doi.org/​10.1017/​9781108499996

[3] Heinz-Peter Breuer and Francesco Petruccione. The theory of open quantum systems. Oxford University Press, 2002. doi:10.1093/​acprof:oso/​9780199213900.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[4] Victor V. Albert and Liang Jiang. Symmetries and conserved quantities in Lindblad master equations. Phys. Rev. A, 89:022118, 2014. doi:10.1103/​PhysRevA.89.022118.
https:/​/​doi.org/​10.1103/​PhysRevA.89.022118

[5] Bernhard Baumgartner and Heide Narnhofer. Analysis of quantum semigroups with GKS–lindblad generators: II. General. Journal of Physics A: Mathematical and Theoretical, 41(39):395303, 2008. doi:10.1088/​1751-8113/​41/​39/​395303.
https:/​/​doi.org/​10.1088/​1751-8113/​41/​39/​395303

[6] Victor V. Albert. Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states. Quantum, 3:151, 2019. doi:10.22331/​q-2019-06-06-151.
https:/​/​doi.org/​10.22331/​q-2019-06-06-151

[7] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91:025001, 2019. doi:10.1103/​RevModPhys.91.025001.
https:/​/​doi.org/​10.1103/​RevModPhys.91.025001

[8] Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys., 79:555–609, 2007. doi:10.1103/​RevModPhys.79.555.
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[9] Iman Marvian and Robert W. Spekkens. The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations. New Journal of Physics, 15(3):033001, 2013. doi:10.1088/​1367-2630/​15/​3/​033001.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​3/​033001

[10] Iman Marvian and Robert W. Spekkens. Extending Noether's theorem by quantifying the asymmetry of quantum states. Nature communications, 5:3821, 2014. doi:10.1038/​ncomms4821.
https:/​/​doi.org/​10.1038/​ncomms4821

[11] Matteo Lostaglio, Kamil Korzekwa, and Antony Milne. Markovian evolution of quantum coherence under symmetric dynamics. Phys. Rev. A, 96:032109, 2017. doi:10.1103/​PhysRevA.96.032109.
https:/​/​doi.org/​10.1103/​PhysRevA.96.032109

[12] Berislav Buča and Tomaž Prosen. A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains. New Journal of Physics, 14(7):073007, 2012. doi:10.1088/​1367-2630/​14/​7/​073007.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​7/​073007

[13] Edward Witten. A mini-introduction to information theory. arXiv:1805.11965.
https:/​/​doi.org/​10.1007/​s40766-020-00004-5
arXiv:1805.11965

[14] Mark M. Wilde. Quantum information theory. Cambridge University Press, 2013. doi:10.1017/​CBO9781139525343.
https:/​/​doi.org/​10.1017/​CBO9781139525343

[15] Kristan Temme, Michael James Kastoryano, Mary Beth Ruskai, Michael Marc Wolf, and Frank Verstraete. The $\chi^2$-divergence and mixing times of quantum Markov processes. Journal of Mathematical Physics, 51(12):122201, 2010. doi:10.1063/​1.3511335.
https:/​/​doi.org/​10.1063/​1.3511335

[16] Michael J. Kastoryano, David Reeb, and Michael M. Wolf. A cutoff phenomenon for quantum Markov chains. Journal of Physics A: Mathematical and Theoretical, 45(7):075307, 2012. doi:10.1088/​1751-8113/​45/​7/​075307.
https:/​/​doi.org/​10.1088/​1751-8113/​45/​7/​075307

[17] Michael J. Kastoryano and Kristan Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. Journal of Mathematical Physics, 54(5):052202, 2013. doi:10.1063/​1.4804995.
https:/​/​doi.org/​10.1063/​1.4804995

[18] Frank Hansen. Metric adjusted skew information. Proceedings of the National Academy of Sciences, 105(29):9909–9916, 2008. doi:10.1073/​pnas.0803323105.
https:/​/​doi.org/​10.1073/​pnas.0803323105

[19] E. A. Morozova and N. N. Chentsov. Markov invariant geometry on manifolds of states. Journal of Soviet Mathematics, 56(5):2648–2669, 1991. doi:10.1007/​BF01095975.
https:/​/​doi.org/​10.1007/​BF01095975

[20] Dénes Petz. Monotone metrics on matrix spaces. Linear Algebra and its Applications, 244:81 – 96, 1996. doi:10.1016/​0024-3795(94)00211-8.
https:/​/​doi.org/​10.1016/​0024-3795(94)00211-8

[21] Dénes Petz and Csaba Sudár. Geometries of quantum states. Journal of Mathematical Physics, 37(6):2662–2673, 1996. doi:10.1063/​1.531535.
https:/​/​doi.org/​10.1063/​1.531535

[22] Dénes Petz and Mary Beth Ruskai. Contraction of generalized relative entropy under stochastic mappings on matrices. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(01):83–89, 1998. doi:10.1142/​S0219025798000077.
https:/​/​doi.org/​10.1142/​S0219025798000077

[23] Ingemar Bengtsson and Karol Życzkowski. Geometry of quantum states: An introduction to quantum entanglement. Cambridge University Press, 2017. doi:10.1017/​CBO9780511535048.
https:/​/​doi.org/​10.1017/​CBO9780511535048

[24] Dénes Petz. Quasi-entropies for finite quantum systems. Reports on mathematical physics, 23(1):57–65, 1986. doi:10.1016/​0034-4877(86)90067-4.
https:/​/​doi.org/​10.1016/​0034-4877(86)90067-4

[25] Andrew Lesniewski and Mary Beth Ruskai. Monotone Riemannian metrics and relative entropy on noncommutative probability spaces. Journal of Mathematical Physics, 40(11):5702–5724, 1999. doi:10.1063/​1.533053.
https:/​/​doi.org/​10.1063/​1.533053

[26] Martin Idel. On the structure of positive maps. Master's thesis, Technical University of Munich, 2013.

[27] Jaroslav Novotnỳ, Jiří Maryška, and Igor Jex. Quantum Markov processes: From attractor structure to explicit forms of asymptotic states. The European Physical Journal Plus, 133(8):310, 2018. doi:10.1140/​epjp/​i2018-12109-8.
https:/​/​doi.org/​10.1140/​epjp/​i2018-12109-8

[28] Rajendra Bhatia. Matrix analysis, volume 169. Springer-Verlag, 2013. doi:10.1007/​978-1-4612-0653-8.
https:/​/​doi.org/​10.1007/​978-1-4612-0653-8

[29] Emanuel Knill, Raymond Laflamme, and Lorenza Viola. Theory of quantum error correction for general noise. Phys. Rev. Lett., 84:2525–2528, 2000. doi:10.1103/​PhysRevLett.84.2525.
https:/​/​doi.org/​10.1103/​PhysRevLett.84.2525

[30] Paolo Zanardi. Stabilizing quantum information. Phys. Rev. A, 63:012301, 2000. doi:10.1103/​PhysRevA.63.012301.
https:/​/​doi.org/​10.1103/​PhysRevA.63.012301

[31] David W. Kribs. Quantum channels, wavelets, dilations and representations of $\mathcal{O}_{n}$. Proceedings of the Edinburgh Mathematical Society, 46(2):421–433, 2003. doi:10.1017/​S0013091501000980.
https:/​/​doi.org/​10.1017/​S0013091501000980

[32] E. Brian Davies. Markovian master equations. Communications in mathematical Physics, 39(2):91–110, 1974. doi:10.1007/​BF01608389.
https:/​/​doi.org/​10.1007/​BF01608389

[33] Robert Alicki and Karl Lendi. Quantum dynamical semigroups and applications, volume 717. Springer-Verlag, 2007. doi:10.1007/​3-540-70861-8.
https:/​/​doi.org/​10.1007/​3-540-70861-8

[34] Wojciech Roga, Mark Fannes, and Karol Życzkowski. Davies maps for qubits and qutrits. Reports on Mathematical Physics, 66(3):311–329, 2010. doi:10.1016/​S0034-4877(11)00003-6.
https:/​/​doi.org/​10.1016/​S0034-4877(11)00003-6

[35] Angel Rivas and Susana F. Huelga. Open quantum systems. Springer-Verlag, 2012. doi:10.1007/​978-3-642-23354-8.
https:/​/​doi.org/​10.1007/​978-3-642-23354-8

[36] Lorenzo Campos Venuti and Paolo Zanardi. Dynamical response theory for driven-dissipative quantum systems. Phys. Rev. A, 93:032101, 2016. doi:10.1103/​PhysRevA.93.032101.
https:/​/​doi.org/​10.1103/​PhysRevA.93.032101

[37] L. Lorne Campbell. An extended Čencov characterization of the information metric. Proceedings of the American Mathematical Society, 98(1):135–141, 1986. doi:10.1090/​S0002-9939-1986-0848890-5.
https:/​/​doi.org/​10.1090/​S0002-9939-1986-0848890-5

### Cited by

[1] Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato, Saverio Pascazio, and Kazuya Yuasa, "Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems", Physical Review Letters 126 15, 150401 (2021).

[2] Gerhard Dorn, Enrico Arrigoni, and Wolfgang von der Linden, "Efficient energy resolved quantum master equation for transport calculations in large strongly correlated systems", Journal of Physics A: Mathematical and Theoretical 54 7, 075301 (2021).

[3] Marco Cattaneo, Gian Luca Giorgi, Sabrina Maniscalco, and Roberta Zambrini, "Symmetry and block structure of the Liouvillian superoperator in partial secular approximation", Physical Review A 101 4, 042108 (2020).

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