Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

Victor V. Albert

Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California, USA
Yale Quantum Institute, Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut, USA

This work derives an analytical formula for the asymptotic state---the quantum state resulting from an infinite number of applications of a general quantum channel on some initial state. For channels admitting multiple fixed or rotating points, conserved quantities---the left fixed/rotating points of the channel---determine the dependence of the asymptotic state on the initial state. The formula stems from a Noether-like theorem stating that, for any channel admitting a full-rank fixed point, conserved quantities commute with that channel’s Kraus operators up to a phase. The formula is applied to adiabatic transport of the fixed-point space of channels, revealing cases where the dissipative/spectral gap can close during any segment of the adiabatic path. The formula is also applied to calculate expectation values of noninjective matrix product states (MPS) in the thermodynamic limit, revealing that those expectation values can also be calculated using an MPS with reduced bond dimension and a modified boundary.

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[1] Filippo Caruso, Vittorio Giovannetti, Cosmo Lupo, and Stefano Mancini. Quantum channels and memory effects. Rev. Mod. Phys., 86 (4): 1203-1259, dec 2014. ISSN 0034-6861. 10.1103/​RevModPhys.86.1203.
https:/​/​doi.org/​10.1103/​RevModPhys.86.1203

[2] M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains. Commun. Math. Phys., 144 (3): 443-490, mar 1992. ISSN 0010-3616. 10.1007/​BF02099178.
https:/​/​doi.org/​10.1007/​BF02099178

[3] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix Product State Representations. aug . URL http:/​/​arxiv.org/​abs/​quant-ph/​0608197.
arXiv:quant-ph/0608197

[4] V. Giovannetti, S. Montangero, and Rosario Fazio. Quantum Multiscale Entanglement Renormalization Ansatz Channels. Phys. Rev. Lett., 101 (18): 180503, oct 2008. ISSN 0031-9007. 10.1103/​PhysRevLett.101.180503.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.180503

[5] Robert N. C. Pfeifer, Glen Evenbly, and Guifré Vidal. Entanglement renormalization, scale invariance, and quantum criticality. Phys. Rev. A, 79 (4): 040301, apr 2009. ISSN 1050-2947. 10.1103/​PhysRevA.79.040301.
https:/​/​doi.org/​10.1103/​PhysRevA.79.040301

[6] Scott Aaronson, Mohammad Bavarian, and Giulio Gueltrini. Computability Theory of Closed Timelike Curves. sep 2016. URL http:/​/​arxiv.org/​abs/​1609.05507.
arXiv:1609.05507

[7] Ji-Woong Lee and Shashi Phoha. Operator sum representation for Markov transition models of human inference processes. In 2016 Am. Control Conf., pages 1590-1595. IEEE, jul 2016. ISBN 978-1-4673-8682-1. 10.1109/​ACC.2016.7525143.
https:/​/​doi.org/​10.1109/​ACC.2016.7525143

[8] Victor V. Albert and Liang Jiang. Symmetries and conserved quantities in Lindblad master equations. Phys. Rev. A, 89 (2): 022118, feb 2014. ISSN 1050-2947. 10.1103/​PhysRevA.89.022118.
https:/​/​doi.org/​10.1103/​PhysRevA.89.022118

[9] John Dengis, Robert Konig, and Fernando Pastawski. An optimal dissipative encoder for the toric code. New J. Phys., 16 (1): 013023, jan 2014. ISSN 1367-2630. 10.1088/​1367-2630/​16/​1/​013023.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​1/​013023

[10] Seth Lloyd and Lorenza Viola. Engineering quantum dynamics. Phys. Rev. A, 65 (1): 010101, dec 2001. ISSN 1050-2947. 10.1103/​PhysRevA.65.010101.
https:/​/​doi.org/​10.1103/​PhysRevA.65.010101

[11] B. Kraus, Hans Peter Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller. Preparation of entangled states by quantum Markov processes. Phys. Rev. A, 78 (4): 042307, oct 2008. ISSN 1050-2947. 10.1103/​PhysRevA.78.042307.
https:/​/​doi.org/​10.1103/​PhysRevA.78.042307

[12] J. F. Poyatos, J Ignacio Cirac, and P. Zoller. Quantum Reservoir Engineering with Laser Cooled Trapped Ions. Phys. Rev. Lett., 77 (23): 4728-4731, dec 1996. ISSN 0031-9007. 10.1103/​PhysRevLett.77.4728.
https:/​/​doi.org/​10.1103/​PhysRevLett.77.4728

[13] P. Schindler, Markus Müller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt. Quantum simulation of dynamical maps with trapped ions. Nat. Phys., 9 (6): 361-367, may 2013. ISSN 1745-2473. 10.1038/​nphys2630.
https:/​/​doi.org/​10.1038/​nphys2630

[14] Shi-Jie Wei, Tao Xin, and Gui-Lu Long. Efficient universal quantum channel simulation in IBM's cloud quantum computer. Sci. China Physics, Mech. Astron., 61 (7): 70311, jul 2018. ISSN 1674-7348. 10.1007/​s11433-017-9181-9.
https:/​/​doi.org/​10.1007/​s11433-017-9181-9

[15] Markus Müller, Sebastian Diehl, Guido Pupillo, and Peter Zoller. Engineered Open Systems and Quantum Simulations with Atoms and Ions. In Paul Berman, Ennio Arimondo, and Chun Lin, editors, Adv. Atom. Mol. Opt. Phy. 61, pages 1-80. Academic Press, 2012. ISBN 0123964822. 10.1016/​B978-0-12-396482-3.00001-6.
https:/​/​doi.org/​10.1016/​B978-0-12-396482-3.00001-6

[16] Chao Shen, Kyungjoo Noh, Victor V. Albert, Stefan Krastanov, Michel H. Devoret, Robert J. Schoelkopf, S. M. Girvin, and Liang Jiang. Quantum channel construction with circuit quantum electrodynamics. Phys. Rev. B, 95 (13): 134501, apr 2017. ISSN 2469-9950. 10.1103/​PhysRevB.95.134501.
https:/​/​doi.org/​10.1103/​PhysRevB.95.134501

[17] Francesco Ticozzi and Lorenza Viola. Quantum and classical resources for unitary design of open-system evolutions. Quantum Sci. Technol., 2 (3): 034001, sep 2017. ISSN 2058-9565. 10.1088/​2058-9565/​aa722a.
https:/​/​doi.org/​10.1088/​2058-9565/​aa722a

[18] A. A. Belavin, B. Ya. Zel'dovich, A. M. Perelomov, and V. S. Popov. Relaxation of Quantum Systems with Equidistant Spectra. Sov. Phys. JETP-USSR, 29 (1): 145, 1969. URL http:/​/​www.jetp.ac.ru/​cgi-bin/​e/​index/​e/​29/​1/​p145?a=list.
http:/​/​www.jetp.ac.ru/​cgi-bin/​e/​index/​e/​29/​1/​p145?a=list

[19] G. Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys., 48 (2): 119-130, jun 1976. ISSN 0010-3616. 10.1007/​BF01608499.
https:/​/​doi.org/​10.1007/​BF01608499

[20] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. J. Math. Phys., 17: 821, may 1976. ISSN 00222488. 10.1063/​1.522979.
https:/​/​doi.org/​10.1063/​1.522979

[21] Thomas Banks, Leonard Susskind, and Michael E. Peskin. Difficulties for the evolution of pure states into mixed states. Nucl. Phys. B, 244: 125-134, sep 1984. ISSN 05503213. 10.1016/​0550-3213(84)90184-6.
https:/​/​doi.org/​10.1016/​0550-3213(84)90184-6

[22] Michael M. Wolf and David Perez-Garcia. The inverse eigenvalue problem for quantum channels. 2010. URL http:/​/​arxiv.org/​abs/​1005.4545.
arXiv:1005.4545

[23] Michael M. Wolf. Quantum Channels & Operations Guided Tour, 2010. URL http:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf.
http:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf

[24] Robin Blume-Kohout, Hui Khoon Ng, David Poulin, and Lorenza Viola. Information-preserving structures: A general framework for quantum zero-error information. Phys. Rev. A, 82 (6): 062306, dec 2010. ISSN 1050-2947. 10.1103/​PhysRevA.82.062306.
https:/​/​doi.org/​10.1103/​PhysRevA.82.062306

[25] Ji Guan, Yuan Feng, and Mingsheng Ying. The Structure of Decoherence-free Subsystems. feb 2018a. URL http:/​/​arxiv.org/​abs/​1802.04904.
arXiv:1802.04904

[26] J. Novotny, G. Alber, and I. Jex. Asymptotic properties of quantum Markov chains. J. Phys. A Math. Theor., 45 (48): 485301, dec 2012. ISSN 1751-8113. 10.1088/​1751-8113/​45/​48/​485301.
https:/​/​doi.org/​10.1088/​1751-8113/​45/​48/​485301

[27] D Burgarth, G Chiribella, V Giovannetti, P Perinotti, and K Yuasa. Ergodic and mixing quantum channels in finite dimensions. New J. Phys., 15 (7): 073045, jul 2013. ISSN 1367-2630. 10.1088/​1367-2630/​15/​7/​073045.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​7/​073045

[28] J. Novotný, J. Maryška, and Igor Jex. Quantum Markov processes: From attractor structure to explicit forms of asymptotic states. Eur. Phys. J. Plus, 133 (8): 310, aug 2018. ISSN 2190-5444. 10.1140/​epjp/​i2018-12109-8.
https:/​/​doi.org/​10.1140/​epjp/​i2018-12109-8

[29] Ulrich Groh. The peripheral point spectrum of schwarz operators onC *-algebras. Math. Zeitschrift, 176 (3): 311-318, sep 1981. ISSN 0025-5874. 10.1007/​BF01214608.
https:/​/​doi.org/​10.1007/​BF01214608

[30] Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen. Quantum Information Meets Quantum Matter. Quantum Science and Technology. Springer New York, New York, NY, aug 2019. ISBN 978-1-4939-9082-5. 10.1007/​978-1-4939-9084-9.
https:/​/​doi.org/​10.1007/​978-1-4939-9084-9

[31] Sven Bachmann and Bruno Nachtergaele. Product vacua with boundary states. Phys. Rev. B, 86 (3): 035149, jul 2012. ISSN 1098-0121. 10.1103/​PhysRevB.86.035149.
https:/​/​doi.org/​10.1103/​PhysRevB.86.035149

[32] Sven Bachmann and Bruno Nachtergaele. Product Vacua with Boundary States and the Classification of Gapped Phases. Commun. Math. Phys., 329 (2): 509-544, jul 2014. ISSN 0010-3616. 10.1007/​s00220-014-2025-x.
https:/​/​doi.org/​10.1007/​s00220-014-2025-x

[33] D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac. String Order and Symmetries in Quantum Spin Lattices. Phys. Rev. Lett., 100 (16): 167202, apr 2008. ISSN 0031-9007. 10.1103/​PhysRevLett.100.167202.
https:/​/​doi.org/​10.1103/​PhysRevLett.100.167202

[34] Norbert Schuch, Ignacio Cirac, and David Pérez-García. PEPS as ground states: Degeneracy and topology. Ann. Phys. (N. Y)., 325 (10): 2153-2192, oct 2010. ISSN 00034916. 10.1016/​j.aop.2010.05.008.
https:/​/​doi.org/​10.1016/​j.aop.2010.05.008

[35] Nick Bultinck, Dominic J. Williamson, Jutho Haegeman, and Frank Verstraete. Fermionic matrix product states and one-dimensional topological phases. Phys. Rev. B, 95 (7): 075108, feb 2017. ISSN 2469-9950. 10.1103/​PhysRevB.95.075108.
https:/​/​doi.org/​10.1103/​PhysRevB.95.075108

[36] Jutho Haegeman, Michaël Mariën, Tobias J. Osborne, and Frank Verstraete. Geometry of matrix product states: Metric, parallel transport, and curvature. J. Math. Phys., 55 (2): 021902, feb 2014. ISSN 0022-2488. 10.1063/​1.4862851.
https:/​/​doi.org/​10.1063/​1.4862851

[37] Gemma De las Cuevas, J Ignacio Cirac, Norbert Schuch, and David Perez-Garcia. Irreducible forms of matrix product states: Theory and applications. J. Math. Phys., 58 (12): 121901, dec 2017. ISSN 0022-2488. 10.1063/​1.5000784.
https:/​/​doi.org/​10.1063/​1.5000784

[38] Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Tangent-space methods for uniform matrix product states. SciPost Phys. Lect. Notes, page 7, jan 2019. ISSN 2590-1990. 10.21468/​SciPostPhysLectNotes.7.
https:/​/​doi.org/​10.21468/​SciPostPhysLectNotes.7

[39] E. C. G. Sudarshan, P. M. Mathews, and Jayaseetha Rau. Stochastic Dynamics of Quantum-Mechanical Systems. Phys. Rev., 121 (3): 920-924, feb 1961. ISSN 0031-899X. 10.1103/​PhysRev.121.920.
https:/​/​doi.org/​10.1103/​PhysRev.121.920

[40] K Kraus. General state changes in quantum theory. Ann. Phys., 64 (2): 311-335, jun 1971. ISSN 00034916. 10.1016/​0003-4916(71)90108-4.
https:/​/​doi.org/​10.1016/​0003-4916(71)90108-4

[41] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra Appl., 10 (3): 285-290, jun 1975. ISSN 00243795. 10.1016/​0024-3795(75)90075-0.
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

[42] Raffaella Carbone and Anna Jenčová. On period, cycles and fixed points of a quantum channel. may 2019. URL http:/​/​arxiv.org/​abs/​1905.00857.
arXiv:1905.00857

[43] Carlton M. Caves. Quantum Error Correction and Reversible Operations. J. Supercond., 12 (6): 707-718, 1999. ISSN 08961107. 10.1023/​A:1007720606911.
https:/​/​doi.org/​10.1023/​A:1007720606911

[44] Göran Lindblad. A General No-Cloning Theorem. Lett. Math. Phys., 47 (2): 189-196, 1999. ISSN 03779017. 10.1023/​A:1007581027660.
https:/​/​doi.org/​10.1023/​A:1007581027660

[45] Robin Blume-Kohout, Hui Khoon Ng, David Poulin, and Lorenza Viola. Characterizing the structure of preserved information in quantum processes. Phys. Rev. Lett., 100 (3): 030501, jan 2008. ISSN 0031-9007. 10.1103/​PhysRevLett.100.030501.
https:/​/​doi.org/​10.1103/​PhysRevLett.100.030501

[46] Bernhard Baumgartner and Heide Narnhofer. The Structures of State Space Concerning Quantum Dynamical Semigroups. Rev. Math. Phys., 24: 1250001, mar 2012. ISSN 0129-055X. 10.1142/​S0129055X12500018.
https:/​/​doi.org/​10.1142/​S0129055X12500018

[47] Raffaella Carbone and Yan Pautrat. Irreducible Decompositions and Stationary States of Quantum Channels. Rep. Math. Phys., 77 (3): 293-313, 2016. ISSN 00344877. 10.1016/​S0034-4877(16)30032-5.
https:/​/​doi.org/​10.1016/​S0034-4877(16)30032-5

[48] David E. Evans and Raphael Hoegh-Krohn. Spectral Properties of Positive Maps on C*-Algebras. J. London Math. Soc., s2-17 (2): 345-355, apr 1978. ISSN 00246107. 10.1112/​jlms/​s2-17.2.345.
https:/​/​doi.org/​10.1112/​jlms/​s2-17.2.345

[49] B. V. Rajarama Bhat, Robin Hillier, Nirupama Mallick, and Vijaya Kumar U. Roots of Completely Positive Maps. dec 2018. URL http:/​/​arxiv.org/​abs/​1812.08123.
arXiv:1812.08123

[50] Karl Alicki and Robert Lendi. Quantum Dynamical Semigroups and Applications, volume 717 of Lecture Notes in Physics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007. ISBN 978-3-540-70860-5. 10.1007/​3-540-70861-8.
https:/​/​doi.org/​10.1007/​3-540-70861-8

[51] S. G. Schirmer and Xiaoting Wang. Stabilizing open quantum systems by Markovian reservoir engineering. Phys. Rev. A, 81: 062306, jun 2010. 10.1103/​PhysRevA.81.062306.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062306

[52] Matteo Ippoliti, Leonardo Mazza, Matteo Rizzi, and Vittorio Giovannetti. Perturbative approach to continuous-time quantum error correction. Phys. Rev. A, 91 (4): 042322, apr 2015. ISSN 1050-2947. 10.1103/​PhysRevA.91.042322.
https:/​/​doi.org/​10.1103/​PhysRevA.91.042322

[53] V. I. Oseledets. Completely positive linear mappings, non-Hamiltonian evolution, and quantum stochastic processes. J. Sov. Math., 25 (6): 1529-1557, jun 1984. ISSN 0090-4104. 10.1007/​BF01101650.
https:/​/​doi.org/​10.1007/​BF01101650

[54] Maxim Raginsky. Strictly contractive quantum channels and physically realizable quantum computers. Phys. Rev. A, 65 (3): 032306, feb 2002a. ISSN 1050-2947. 10.1103/​PhysRevA.65.032306.
https:/​/​doi.org/​10.1103/​PhysRevA.65.032306

[55] Maxim Raginsky. Dynamical Aspects of Information Storage in Quantum-Mechanical Systems. PhD thesis, Northwestern University, jul 2002b. URL http:/​/​arxiv.org/​abs/​quant-ph/​0207162.
arXiv:quant-ph/0207162

[56] Daniel Burgarth and Vittorio Giovannetti. The generalized Lyapunov theorem and its application to quantum channels. New J. Phys., 9 (5): 150-150, may 2007. ISSN 1367-2630. 10.1088/​1367-2630/​9/​5/​150.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​5/​150

[57] E. B. Davies. Quantum stochastic processes II. Commun. Math. Phys., 19 (2): 83-105, jun 1970. ISSN 0010-3616. 10.1007/​BF01646628.
https:/​/​doi.org/​10.1007/​BF01646628

[58] Mikel Sanz, David Perez-Garcia, Michael M. Wolf, and Juan I. Cirac. A Quantum Version of Wielandt's Inequality. IEEE Trans. Inf. Theory, 56 (9): 4668-4673, sep 2010. ISSN 0018-9448. 10.1109/​TIT.2010.2054552.
https:/​/​doi.org/​10.1109/​TIT.2010.2054552

[59] Victor V. Albert, Barry Bradlyn, Martin Fraas, and Liang Jiang. Geometry and Response of Lindbladians. Phys. Rev. X, 6 (4): 041031, nov 2016. ISSN 2160-3308. 10.1103/​PhysRevX.6.041031.
https:/​/​doi.org/​10.1103/​PhysRevX.6.041031

[60] Ji Guan, Yuan Feng, and Mingsheng Ying. Decomposition of quantum Markov chains and its applications. J. Comput. Syst. Sci., 95: 55-68, aug 2018b. ISSN 00220000. 10.1016/​j.jcss.2018.01.005.
https:/​/​doi.org/​10.1016/​j.jcss.2018.01.005

[61] David W. Kribs. Quantum channels, wavelets, dilations and representations of On. Proc. Edinburgh Math. Soc., 46 (2): S0013091501000980, jun 2003. ISSN 00130915. 10.1017/​S0013091501000980.
https:/​/​doi.org/​10.1017/​S0013091501000980

[62] Man-Duen Choi and David W. Kribs. Method to Find Quantum Noiseless Subsystems. Phys. Rev. Lett., 96 (5): 050501, feb 2006. ISSN 0031-9007. 10.1103/​PhysRevLett.96.050501.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.050501

[63] Aurelian Gheondea. On Propagation of Fixed Points of Quantum Operations and Beyond. nov 2016. URL http:/​/​arxiv.org/​abs/​1611.04742.
arXiv:1611.04742

[64] Aurelian Gheondea. Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points. Ann. Henri Poincare, 19 (6): 1787-1816, jun 2018. ISSN 1424-0637. 10.1007/​s00023-018-0666-6.
https:/​/​doi.org/​10.1007/​s00023-018-0666-6

[65] Oleg Szehr and Michael M. Wolf. Connected components of irreducible maps and 1D quantum phases. J. Math. Phys., 57 (8): 081901, aug 2016. ISSN 0022-2488. 10.1063/​1.4960557.
https:/​/​doi.org/​10.1063/​1.4960557

[66] Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims. Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Commun. Math. Phys., 309 (3): 835-871, feb 2012. ISSN 0010-3616. 10.1007/​s00220-011-1380-0.
https:/​/​doi.org/​10.1007/​s00220-011-1380-0

[67] John E. Gough, Tudor S. Ratiu, and Oleg G. Smolyanov. Noether's theorem for dissipative quantum dynamical semi-groups. J. Math. Phys., 56 (2): 022108, feb 2015. ISSN 0022-2488. 10.1063/​1.4907985.
https:/​/​doi.org/​10.1063/​1.4907985

[68] Victor V. Albert. Lindbladians with multiple steady states: theory and applications. PhD thesis, Yale University, 2017. URL https:/​/​arxiv.org/​abs/​1802.00010.
arXiv:1802.00010

[69] Iman Marvian and Robert W Spekkens. Extending Noether's theorem by quantifying the asymmetry of quantum states. Nat. Commun., 5 (1): 3821, sep 2014. ISSN 2041-1723. 10.1038/​ncomms4821.
https:/​/​doi.org/​10.1038/​ncomms4821

[70] Shenggang Ying, Yuan Feng, Nengkun Yu, and Mingsheng Ying. Reachability Probabilities of Quantum Markov Chains. In Pedro R. D'Argenio and Hernán Melgratti, editors, CONCUR 2013 - Concurr. Theory, pages 334-348. Springer Berlin Heidelberg, 2013. 10.1007/​978-3-642-40184-8_24.
https:/​/​doi.org/​10.1007/​978-3-642-40184-8_24

[71] Giuseppe Ilario Cirillo and Francesco Ticozzi. Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups. J. Phys. A Math. Theor., 48 (8): 085302, feb 2015. ISSN 1751-8113. 10.1088/​1751-8113/​48/​8/​085302.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​085302

[72] J. G. Kemeny and J. L. Snell. Finite Markov chains. Springer-Verlag, New York, 2nd edition, 1983. URL https:/​/​www.springer.com/​us/​book/​9780387901923.
https:/​/​www.springer.com/​us/​book/​9780387901923

[73] Daniel A. Lidar, Isaac L. Chuang, and K. Birgitta Whaley. Decoherence-Free Subspaces for Quantum Computation. Phys. Rev. Lett., 81: 2594, sep 1998. ISSN 0031-9007. 10.1103/​PhysRevLett.81.2594.
https:/​/​doi.org/​10.1103/​PhysRevLett.81.2594

[74] Daniel A. Lidar and K. Birgitta Whaley. Decoherence-Free Subspaces and Subsystems. In Fabio Benatti and Roberto Floreanini, editors, Irreversible Quantum Dyn., volume 622 of Lecture Notes in Physics, chapter 5, page 83. Springer, Berlin, Heidelberg, jun 2003. ISBN 978-3-540-40223-7. 10.1007/​3-540-44874-8.
https:/​/​doi.org/​10.1007/​3-540-44874-8

[75] Raisa I. Karasik, Karl-Peter Marzlin, Barry C. Sanders, and K. Birgitta Whaley. Criteria for dynamically stable decoherence-free subspaces and incoherently generated coherences. Phys. Rev. A, 77 (5): 052301, may 2008. ISSN 1050-2947. 10.1103/​PhysRevA.77.052301.
https:/​/​doi.org/​10.1103/​PhysRevA.77.052301

[76] Takeo Kamizawa. Algebraic Method in the Analysis of Decoherence-Free Subspaces in Open Quantum Systems. Int. J. Theor. Phys., pages 1-13, jan 2018. 10.1007/​s10773-017-3657-3.
https:/​/​doi.org/​10.1007/​s10773-017-3657-3

[77] Emanuel Knill, Raymond Laflamme, and Lorenza Viola. Theory of Quantum Error Correction for General Noise. Phys. Rev. Lett., 84: 2525-2528, mar 2000. ISSN 0031-9007. 10.1103/​PhysRevLett.84.2525.
https:/​/​doi.org/​10.1103/​PhysRevLett.84.2525

[78] Mizanur Rahaman. Multiplicative properties of quantum channels. J. Phys. A Math. Theor., 50 (34): 345302, aug 2017. ISSN 1751-8113. 10.1088/​1751-8121/​aa7b57.
https:/​/​doi.org/​10.1088/​1751-8121/​aa7b57

[79] John A. Holbrook, David W. Kribs, and Raymond Laflamme. Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction. Quantum Inf. Process., 2 (5): 381-419, oct 2003. ISSN 1570-0755. 10.1023/​B:QINP.0000022737.53723.b4.
https:/​/​doi.org/​10.1023/​B:QINP.0000022737.53723.b4

[80] E. Knill. Protected realizations of quantum information. Phys. Rev. A, 74 (4): 042301, oct 2006. ISSN 1050-2947. 10.1103/​PhysRevA.74.042301.
https:/​/​doi.org/​10.1103/​PhysRevA.74.042301

[81] Takanori Maehara and Kazuo Murota. A numerical algorithm for block-diagonal decomposition of matrix *-algebras with general irreducible components. Jpn. J. Ind. Appl. Math., 27 (2): 263-293, sep 2010. ISSN 0916-7005. 10.1007/​s13160-010-0007-8.
https:/​/​doi.org/​10.1007/​s13160-010-0007-8

[82] Xiaoting Wang, Mark Byrd, and Kurt Jacobs. Numerical method for finding decoherence-free subspaces and its applications. Phys. Rev. A, 87 (1): 012338, jan 2013. ISSN 1050-2947. 10.1103/​PhysRevA.87.012338.
https:/​/​doi.org/​10.1103/​PhysRevA.87.012338

[83] Ion Nechita and Clément Pellegrini. Random repeated quantum interactions and random invariant states. Probab. Theory Relat. Fields, 152 (1-2): 299-320, feb 2012. ISSN 0178-8051. 10.1007/​s00440-010-0323-6.
https:/​/​doi.org/​10.1007/​s00440-010-0323-6

[84] Laurent Bruneau, Alain Joye, and Marco Merkli. Repeated interactions in open quantum systems. J. Math. Phys., 55 (7): 075204, jul 2014. ISSN 0022-2488. 10.1063/​1.4879240.
https:/​/​doi.org/​10.1063/​1.4879240

[85] M. Born and V. Fock. Beweis des Adiabatensatzes. Zeitschrift Phys., 51 (3-4): 165-180, mar 1928. ISSN 1434-6001. 10.1007/​BF01343193.
https:/​/​doi.org/​10.1007/​BF01343193

[86] Tosio Kato. On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn., 5: 435, 1950. 10.1143/​JPSJ.5.435.
https:/​/​doi.org/​10.1143/​JPSJ.5.435

[87] M. V. Berry. Quantal Phase Factors Accompanying Adiabatic Changes. Proc. R. Soc. A, 392: 45-57, mar 1984. 10.1098/​rspa.1984.0023.
https:/​/​doi.org/​10.1098/​rspa.1984.0023

[88] Frank Wilczek and A. Zee. Appearance of Gauge Structure in Simple Dynamical Systems. Phys. Rev. Lett., 52 (24): 2111, jun 1984. ISSN 0031-9007. 10.1103/​PhysRevLett.52.2111.
https:/​/​doi.org/​10.1103/​PhysRevLett.52.2111

[89] J. E. Avron, M. Fraas, G. M. Graf, and P. Grech. Adiabatic Theorems for Generators of Contracting Evolutions. Commun. Math. Phys., 314 (1): 163-191, may 2012. ISSN 0010-3616. 10.1007/​s00220-012-1504-1.
https:/​/​doi.org/​10.1007/​s00220-012-1504-1

[90] G Nenciu and G Rasche. On the adiabatic theorem for nonself-adjoint Hamiltonians. J. Phys. A Math. Gen., 25 (21): 5741-5751, nov 1992. ISSN 0305-4470. 10.1088/​0305-4470/​25/​21/​027.
https:/​/​doi.org/​10.1088/​0305-4470/​25/​21/​027

[91] M V Berry and R Uzdin. Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon. J. Phys. A Math. Theor., 44 (43): 435303, oct 2011. ISSN 1751-8113. 10.1088/​1751-8113/​44/​43/​435303.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​43/​435303

[92] Raam Uzdin, Alexei Mailybaev, and Nimrod Moiseyev. On the observability and asymmetry of adiabatic state flips generated by exceptional points. J. Phys. A Math. Theor., 44 (43): 435302, oct 2011. ISSN 1751-8113. 10.1088/​1751-8113/​44/​43/​435302.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​43/​435302

[93] J. Holler, N. Read, and J.G.E. Harris. Non-Hermitian adiabatic transport in the space of exceptional points. 2019. URL http:/​/​arxiv.org/​abs/​1809.07175.
arXiv:1809.07175

[94] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics: a Foundation for Computer Science. Addison-Wesley, New York, 2nd edition, 1994. URL https:/​/​www.worldcat.org/​title/​concrete-mathematics-a-foundation-for-computer-science/​oclc/​29357079.
https:/​/​www.worldcat.org/​title/​concrete-mathematics-a-foundation-for-computer-science/​oclc/​29357079

[95] M. S. Sarandy and Daniel A. Lidar. Abelian and non-Abelian geometric phases in adiabatic open quantum systems. Phys. Rev. A, 73 (6): 062101, jun 2006. ISSN 1050-2947. 10.1103/​PhysRevA.73.062101.
https:/​/​doi.org/​10.1103/​PhysRevA.73.062101

[96] Barry Simon. Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase. Phys. Rev. Lett., 51 (24): 2167, dec 1983. ISSN 0031-9007. 10.1103/​PhysRevLett.51.2167.
https:/​/​doi.org/​10.1103/​PhysRevLett.51.2167

[97] Lorenzo Campos Venuti, Tameem Albash, Daniel A. Lidar, and Paolo Zanardi. Adiabaticity in open quantum systems. Phys. Rev. A, 93 (3): 032118, mar 2016. ISSN 2469-9926. 10.1103/​PhysRevA.93.032118.
https:/​/​doi.org/​10.1103/​PhysRevA.93.032118

[98] Hiroshi Ueda, Isao Maruyama, and Kouichi Okunishi. Uniform Matrix Product State in the Thermodynamic Limit. J. Phys. Soc. Jpn., 80 (2): 023001, feb 2011. ISSN 0031-9015. 10.1143/​JPSJ.80.023001.
https:/​/​doi.org/​10.1143/​JPSJ.80.023001

[99] F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M. M. Wolf. Renormalization-Group Transformations on Quantum States. Phys. Rev. Lett., 94 (14): 140601, apr 2005. ISSN 0031-9007. 10.1103/​PhysRevLett.94.140601.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.140601

[100] Tzu-Chieh Wei. Entanglement under the renormalization-group transformations on quantum states and in quantum phase transitions. Phys. Rev. A, 81 (6): 062313, jun 2010. ISSN 1050-2947. 10.1103/​PhysRevA.81.062313.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062313

[101] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete. Matrix product density operators: Renormalization fixed points and boundary theories. Ann. Phys., 378: 100-149, mar 2017. ISSN 00034916. 10.1016/​j.aop.2016.12.030.
https:/​/​doi.org/​10.1016/​j.aop.2016.12.030

[102] J. Ignacio Cirac, Didier Poilblanc, Norbert Schuch, and Frank Verstraete. Entanglement spectrum and boundary theories with projected entangled-pair states. Phys. Rev. B, 83 (24): 245134, 2011. ISSN 1098-0121. 10.1103/​PhysRevB.83.245134.
https:/​/​doi.org/​10.1103/​PhysRevB.83.245134

[103] Michal Bialonczyk, Andrzej Jamiolkowski, and Karol Zyczkowski. Application of Shemesh theorem to quantum channels. J. Math. Phys., 59 (10): 102204, oct 2018. ISSN 0022-2488. 10.1063/​1.5027616.
https:/​/​doi.org/​10.1063/​1.5027616

Cited by

[1] Ji Guan, Yuan Feng, Andrea Turrini, and Mingsheng Ying, "Model Checking Applied to Quantum Physics", arXiv:1902.03218.

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