Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA
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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.
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► 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
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[1] 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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