Abstract
The concept of balance between two state-preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived, and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly explored.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal. 45, 245–273 (1982)
Accardi, L., Fagnola, F., Quezada, R.: Weighted detailed balance and local KMS condition for non-equilibrium stationary states. In: Perspective Nonequilibrium Statistical Physics, vol. 97, pp. 318–356. Bussei Kenkyu (2011)
Accardi, L., Fagnola, F., Quezada, R.: On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19, 1650009 (2016)
Accardi, L., Fidaleo, F.: Bose-Einstein condensation and condensation of \(q\)-particles in equilibrium and nonequilibrium thermodynamics. Rep. Math. Phys. 77, 153–182 (2016)
Accardi, L., Imafuku, K.: Dynamical detailed balance and local KMS condition for non-equilibrium states. Int. J. Mod. Phys. B 18, 435–467 (2004)
Agarwal, G.S.: Open quantum Markovian systems and the microreversibility. Z. Phys. A Hadrons Nucl. 258, 409–422 (1973)
Alicki, R.: On the detailed balance condition for non-Hamiltonian systems. Rep. Math. Phys. 10, 249–258 (1976)
Albeverio, S., Høegh-Krohn, R.: Frobenius theory for positive maps of von Neumann algebras. Commun. Math. Phys. 64, 83–94 (1978)
Arrighi, P., Patricot, C.: On quantum operations as quantum states. Ann. Phys. 311, 26–52 (2004)
Bannon, J.P., Cameron, J., Mukherjee, K.: The modular symmetry of Markov maps. J. Math. Anal. Appl. 439, 701–708 (2016)
Bannon, J.P., Cameron, J., Mukherjee, K.: On noncommutative joinings. Int. Math. Res. Not. https://doi.org/10.1093/imrn/rnx024
Blackadar, B.: Operator algebras: theory of C*-algebras and von Neumann algebras. In: Encyclopaedia of Mathematical Sciences, vol. 122, Springer, Berlin (2006)
Bolaños-Servin, J.R., Quezada, R.: A cycle decomposition and entropy production for circulant quantum Markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16, 1350016 (2013)
Bolaños-Servin, J.R., Quezada, R.: The \(\Theta \)-KMS adjoint and time reversed quantum Markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18, 1550016 (2015)
Carmichael, H.J., Walls, D.F.: Detailed balance in open quantum Markoffian systems. Z. Phys. B Condens. Matter 23, 299–306 (1976)
Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)
Cipriani, F.: Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras. J. Funct. Anal. 147, 259–300 (1997)
Connes, A.: Noncommutative Geometry. Academic Press Inc, San Diego, CA (1994)
de Pillis, J.: Linear transformations which preserve hermitian and positive semidefinite operators. Pac. J. Math. 23, 129–137 (1967)
Dereziński, J., Fruboes, R.: Fermi golden rule and open quantum systems. In: Attal, S., et al. (eds.) Open Quantum Systems III. Lecture Notes in Mathematics, vol. 1882, pp. 67–116. Springer, Berlin (2006)
Dixmier, J.: Von Neumann Algebras. North-Holland, Amsterdam (1981)
Duvenhage, R.: Joinings of W*-dynamical systems. J. Math. Anal. Appl. 343, 175–181 (2008)
Duvenhage, R.: Ergodicity and mixing of W*-dynamical systems in terms of joinings. Ill. J. Math. 54, 543–566 (2010)
Duvenhage, R.: Relatively independent joinings and subsystems of W*-dynamical systems. Stud. Math. 209, 21–41 (2012)
Duvenhage, R., Snyman, M.: Detailed balance and entanglement. J. Phys. A 48, 155303 (2015)
Duvenhage, R., Ströh, A.: Recurrence and ergodicity in unital *-algebras. J. Math. Anal. Appl. 287, 430–443 (2003)
Fagnola, F., Rebolledo, R.: Algebraic conditions for convergence of a quantum Markov semigroup to a steady state. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11, 467–474 (2008)
Fagnola, F., Rebolledo, R.: From classical to quantum entropy production. In: Ouerdiane, H., Barhoumi, A. (eds) Proceedings of the 29th Conference on Quantum Probability and Related Topics, QP–PQ: Quantum Probability and White Noise Analysis, 25, p. 245. World Scientific, Hackensack, NJ (2010)
Fagnola, F., Rebolledo, R.: Entropy production for quantum Markov semigroups. Commun. Math. Phys. 335, 547–570 (2015)
Fagnola, F., Umanità, V.: Generators of KMS symmetric Markov semigroups on B(h) symmetry and quantum detailed balance. Commun. Math. Phys. 298, 523–547 (2010)
Falcone, T.: \(L^{2}\)-von Neumann modules, their relative tensor products and the spatial derivative. Ill. J. Math. 44, 407–437 (2000)
Fellah, D.: Return to thermal equilibrium. Lett. Math. Phys. 80, 101–113 (2007)
Fidaleo, F.: An ergodic theorem for quantum diagonal measures. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12, 307–320 (2009)
Fidaleo, F., Viaggiu, S.: A proposal for the thermodynamics of certain open systems. Physica A 468, 677–690 (2017)
Frigerio, A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys. 63, 269–276 (1978)
Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)
Glasner, E.: Ergodic theory via joinings. In: Mathematical Surveys and Monographs 101. American Mathematical Society, Providence, RI (2003)
Goldstein, S., Lindsay, J.M.: Beurling-Deny conditions for KMS-symmetric dynamical semigroups. C. R. Acad. Sci. Paris Sér. I Math. 317, 1053–1057 (1993)
Goldstein, S., Lindsay, J.M.: KMS-symmetric Markov semigroups. Math. Z. 219, 591–608 (1995)
Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of \(N\)-level systems. J. Math. Phys. 17, 821–825 (1976)
Grabowski, J., Kuś, M., Marmo, G.: On the relation between states and maps in infinite dimensions. Open Syst. Inf. Dyn. 14, 355–370 (2007)
Jamiołkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972)
Jiang, M., Luo, S., Fu, S.: Channel-state duality. Phys. Rev. A 87, 022310 (2013)
Kerr, D., Li, H., Pichot, M.: Turbulence, representations and trace-preserving actions. Proc. Lond. Math. Soc. 3(100), 459–484 (2010)
Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys. 57, 97–110 (1977)
Kümmerer, B., Schwieger, K.: Diagonal couplings of quantum Markov chains. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19, 1650012 (2016)
Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)
Majewski, W.A.: The detailed balance condition in quantum statistical mechanics. J. Math. Phys. 25, 614–616 (1984)
Majewski, W.A.: Return to equilibrium and stability of dynamics (semigroup dynamics case). J. Stat. Phys. 55, 417–433 (1989)
Majewski, W.A., Streater, R.F.: Detailed balance and quantum dynamical maps. J. Phys. A 31, 7981–7995 (1998)
Majewski, W.A.: Quantum dynamical maps and return to equilibrium. Acta Phys. Polon. B 32, 1467–1474 (2001)
Niculescu, C.P., Ströh, A., Zsidó, L.: Noncommutative extensions of classical and multiple recurrence theorems. J. Oper. Theory 50, 3–52 (2003)
Ohya, M., Petz, D.: Quantum entropy and its use. In: Texts and Monographs in Physics. Springer, Berlin (1993)
Pandiscia, C.: Ergodic dilation of a quantum dynamical system. Confluentes Math. 6, 77–91 (2014)
Parthasarathy, K.R.: An introduction to quantum stochastic calculus. In: Monographs in Mathematics, vol. 85. Birkhäuser Verlag, Basel (1992)
Petz, D.: A dual in von Neumann algebras with weights. Q. J. Math. Oxf. Ser. 2(35), 475–483 (1984)
Popa, S.: Correspondences (preliminary version). Unpublished manuscript. http://www.math.ucla.edu/~popa/popa-correspondences.pdf
Rudolph, D.J.: An example of a measure preserving map with minimal self-joinings, and applications. J. Anal. Math. 35, 97–122 (1979)
Sauvageot, J.-L.: Sur le produit tensoriel relatif d’espaces de Hilbert. J. Oper. Theory 9, 237–252 (1983)
Sauvageot, J.-L., Thouvenot, J.-P.: Une nouvelle dé finition de l’entropie dynamique des systèmes non commutatifs. Commun. Math. Phys. 145, 411–423 (1992)
Spohn, H.: Approach to equilibrium for completely positive dynamical semigroups of \(N\)-level systems. Rep. Math. Phys. 10, 189–194 (1976)
Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)
Takesaki, M.: Theory of operator algebras II. In: Encyclopaedia of Mathematical Sciences, vol. 125. Operator Algebras and Non-commutative Geometry, vol. 6. Springer, Berlin (2003)
Verstraete, F., Verschelde, H.: On quantum channels. arXiv:quant-ph/0202124v2
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Claude Alain Pillet.
Rights and permissions
About this article
Cite this article
Duvenhage, R., Snyman, M. Balance Between Quantum Markov Semigroups. Ann. Henri Poincaré 19, 1747–1786 (2018). https://doi.org/10.1007/s00023-018-0664-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-018-0664-8