Journal of Soviet Mathematics

, Volume 25, Issue 6, pp 1529–1557 | Cite as

Completely positive linear mappings, non-Hamiltonian evolution, and quantum stochastic processes

  • V. I. Oseledets


The present survey is mainly devoted to works published from 1969–1980 in Ref. Mat. Zh. in which completely positive linear mappings are studied that arise, in particular, in the quantum theory of open systems, the quantum theory of measurements, and in problems of the dynamics of a small system interacting with a large system. Here the probabilistic aspect is singled out, and analogies and connections with ordinary Markov processes are indicated.


Open System Linear Mapping Stochastic Process Quantum Theory Markov Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    L. Accardi, “On a noncommutative Markov property,” Funkts. Anal. Prilozhen.,9, No. 1, 1–8 (1975).Google Scholar
  2. 2.
    I. V. Aleksandrov, The Theory of Magnetic Relaxation [in Russian], Nauka, Moscow (1975).Google Scholar
  3. 3.
    Yu. N. Barabanenkov, “On the solution of the Liouville stochastic equation for particles in a field of variable scatterers,” Teor. Mat. Fiz.,29, No. 2, 244–254 (1976).Google Scholar
  4. 4.
    Yu. N. Barabanenkov and V. D. Ozrin, “The basic kinetic equation for a particle in a field of randomly varying scatterers,” Teor. Mat. Fiz.,36, No. 2, 240–251 (1978).Google Scholar
  5. 5.
    Yu. N. Barabanenkov, V. D. Ozrin, and A. V. Shelest, “On the theory of stochastic processes in quantum dynamical systems,” Teor. Mat. Fiz.,42, No. 2, 232–242 (1980).Google Scholar
  6. 6.
    D. I. Blokhintsev, Quantum Mechanics: Lectures on Selected Questions [in Russian], Atomizdat, Moscow (1981).Google Scholar
  7. 7.
    N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics [in Russian], Izd. Akad. Nauk SSSR, Kiev (1945).Google Scholar
  8. 8.
    N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946).Google Scholar
  9. 9.
    V. D. Vainshtein, The Problem of Quantum-Mechanical Measurements and Macroscopic Description of Quantum Systems [in Russian], Tomsk Univ. (1981).Google Scholar
  10. 10.
    M. Sh. Gol'dshtein, “On the convergence of conditional mathematical expectations in von Neumann algebras,” UzSSR Fanlar Akad. Dokl., Dokl. Akad. Nauk Uzb. SSR, No. 12, 7–8 (1978).Google Scholar
  11. 11.
    M. Sh. Gol'dshtein, “An individual ergodic theorem for substochastic operators in von Neumann algebras,” UzSSR Fanlar Akad. Dokl., Dokl. Akad. Nauk Uzb. SSR, No. 7, 3–5 (1979).Google Scholar
  12. 12.
    M. Sh. Gol'dshtein, “An individual ergodic theorm for positive linear mappings of von Neumann algebras,” Funkts. Analiz Prilozhen.,14, No. 4, 69–70 (1980).Google Scholar
  13. 13.
    B. A. Grishanin, Quantum Electrodynamics for Radiophysics [in Russian], Moscow State Univ. (1981).Google Scholar
  14. 14.
    B. A. Grishanin, “Markov methods of analysis of stochastic quantum dynamics of quasiclassical open systems,” Teor. Mat. Fiz.,48, No. 3, 396–409 (1981).Google Scholar
  15. 15.
    B. M. Gurevich and V. I. Oseledets, “Some mathematical problems connected with nonequilibrium statistical mechanics of an infinite number of particles,” in: Probability Theory, Mathematical Statistics, Theory of Cybernetics, Vol. 14, (Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR), Moscow (1977), pp. 5–39.Google Scholar
  16. 16.
    J. Klauder and E. Sudarshan, Foundations of Quantum Optics [Russian translation], Mir, Mocow (1970).Google Scholar
  17. 17.
    V. F. Los', “On the dynamics of a subsystem interacting with a thermostat,” Teor. Mat. Fiz.,39, No. 3, 393–402 (1979).Google Scholar
  18. 18.
    M. Lax, Fluctuations and Coherent Phenomena [Russian translation]. Mir, Moscow (1974).Google Scholar
  19. 19.
    V. A. Malyshev and Yu. A. Terletskii, “A limit theorem for noncommutative fields,” Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 3, 47–51 (1978).Google Scholar
  20. 20.
    V. P. Maslov, Complex Markov Chains and Feynman's Path Integral for Nonlinear Equations [in Russian], Nauka, Moscow (1975).Google Scholar
  21. 21.
    E. A. Morozova and N. N. Chentsov, “Probability matrices and stochastic supermatrices,” Inst. Prikl. Mat. Akad. Nauk SSSR, Preprint, No. 84, Moscow (1973).Google Scholar
  22. 22.
    E. A. Morozova and N. N. Chentsov, “The structure of the family of stationary states of a quantum Markov chain,” Inst. Prikl. Mat. Akad. Nauk SSSR, Preprint, No. 130, Moscow (1976).Google Scholar
  23. 23.
    E. A. Morozova and N. N. Chentsov, Elements of Stochastic Quantum Logic [in Russian], Novosibirsk (1977).Google Scholar
  24. 24.
    E. A. Morozova and N. N. Chentsov, “The algebra of bounded harmonic functions of a countable Markov chain (noncommutative theory),” Inst. Prikl. Mat. Akad. Nauk SSSR, Preprint, No. 1 (1980–1981).Google Scholar
  25. 25.
    E. A. Morozova and N. N. Chentsov, “Noncommutative quantum logic (finite-dimensional theory),” Inst, Prikl. Mat. Akad. Nauk SSSR, Preprint, No. 57 (1981).Google Scholar
  26. 26.
    E. A. Morozova and N. N. Chentsov, “Probability distributions on noncommutative logics: finite-dimensioal theory,” Inst. Prikl. Mat. Akad. Nauk SSSR, Preprint, No. 129 (1981).Google Scholar
  27. 27.
    I. P. Pavlotskii and L. G. Shekhovtsova, “The quantum-mechanical kinetic equation for a nonequilibrium particle in a thermostat,” Dokl. Akad. Nauk SSSR,210, No. 6, 1323–1326 (1973).Google Scholar
  28. 28.
    N. V. Trunov and A. N. Sherstnev, “On the general theory of integration in operator algebras relative to weight 1,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 79–88 (1978).Google Scholar
  29. 29.
    K. Hepp, “Results and problems in irreversible statistical mechanics of open systems,” in: Gibbs States and Statistical Physics, Moscow (1978), pp. 241–255.Google Scholar
  30. 30.
    A. S. Kholevo, Probabilistic and Statistical Aspects of Quantum Theory [in Russian], Nauka, Moscow (1980).Google Scholar
  31. 31.
    B. S. Tsirel'son. “On the formal description of quantum systems close to systems of stochastic automata,” in: Interactions of Markov Processes and Their Applications in Biology [in Russian], Pushchino (1979), pp. 100–138.Google Scholar
  32. 32.
    A. N. Sherstnev; “On a noncommutative analog of the space L1,” Usp. Mat. Nauk,33, No. 1, 231–232 (1978).Google Scholar
  33. 33.
    L. Accardi, “Nonrelativistic quantum mechanics as a noncommutative Markov process,” Adv. Math.,20, No. 3, 329–366 (1976).Google Scholar
  34. 34.
    L. Accardi, “Noncommutative Markov chains associated to a preassigned evolution: an application to the quantum theory of measurement,” Adv. Math.,29, No. 2, 226–243 (1979).Google Scholar
  35. 35.
    L. Accardi, “Local perturbations of conditional expectations,” J. Math. Anal. Appl.,72, No. 1, 34–69 (1979).Google Scholar
  36. 36.
    P. M. Alberti, “Fortsetzungssätze fur Markovsche Halbgruppen stochastischer und spurerhaltender linearer Abbildungen über Matrixalgebren,” Wiss. Z. Karl-Marx Univ., Leipzig, Math.-Naturwiss. R.,29, No. 4, 393–400 (1980).Google Scholar
  37. 37.
    S. Albeverio and R. J. Hoegh-Krohn, “Frobenius theory for positive maps of von Neumann algebras,” Preprint Ser. Inst. of Math. Univ. Oslo, No. 10 (1977).Google Scholar
  38. 38.
    S. A. Albeverio and R. J. Hoegh-Krohn, “Frobenius theory for positive maps of von Neumann algebras,” Commun. Math. Phys.,64; No. 1, 83–94 (1978).Google Scholar
  39. 39.
    S. A. Albeverio, R. J. Hoegh-Krohn, and G. Olsen, “Dynamical semigroups and Markov processes on C*-algebras,” J. Reine Angew. Math., No. 319, 25–37 (1980).Google Scholar
  40. 40.
    R. Alicki, “The Markovian master equations for unstable particles,” Repts. Math. Phys.,11, No. 1, 1–6 (1977).Google Scholar
  41. 41.
    R. Alicki, “The Markov master equations and the Fermi golden rule,” Int. J. Theory. Phys.,16, No. 5, 351–355 (1977).Google Scholar
  42. 42.
    R. Alicki, “On the detailed balance condition for non-Hamiltonian systems,” Repts. Math. Phys.,10, No. 2, 249–258 (1976).Google Scholar
  43. 43.
    R. Alicki, “The theory of open systems in application to unstable particles,” Repts. Math. Phys.,14, No. 1, 27–42 (1978).Google Scholar
  44. 44.
    R. Alicki, “On the entropy production for the Davies model of a heat conductions” J. Statist. Phys.,20, No. 6, 671–677 (1979).Google Scholar
  45. 45.
    G. Ananthawrishna, E. C. G. Sudarshan, and V. Gorini, “Entropy increase for a class of dynamical maps,” Repts. Math. Phys.,8, No. 1, 25–32 (1975).Google Scholar
  46. 46.
    W. B. Arveson, “Subalgebras of C*-algebras,” Acta Math.,123, Nos. 3–4, 141–225 (1969).Google Scholar
  47. 47.
    N. L. Balazs, “The Fokker-Planck and Kramers-Chandresekhar equations for semiclassical bosons and fermions,” Physica,A94, Nos. 3–4, 474–480 (1978).Google Scholar
  48. 48.
    C. J. K. Batty, “The strong law of large numbers for states and traces of a W*-algebra,” Z. Wahrscheinlichkeitstheor. Verw. Geb.,48, No. 2, 117–191 (1979).Google Scholar
  49. 49.
    V. P. Belavkin, “Optimization of quantum observables and control,” Lect. Notes Control Inf. Sci.,22, 143–149 (1980).Google Scholar
  50. 50.
    N. N. Bogolyubov, “On the stochastic processes in the dynamical systems,” Soobshch. OIYaI, Dubna, E17-10514 (1977).Google Scholar
  51. 51.
    O. Bratelli and A. Kishimoto, “Generation of semigroups and two-dimensional quantum lattice systems,” J. Funct. Anal.,35, No. 3, 344–368 (1980).Google Scholar
  52. 52.
    E. Buffet and Ph. A. Martin, “Dynamics of the open BCS model,” J. Statist, Phys.,18, No. 6, 585–632.Google Scholar
  53. 53.
    J. Burzlaff, “Canonical quantization of dissipative systems,” Repts. Math. Phys.,16, No. 1, 101–110 (1979).Google Scholar
  54. 54.
    E. Chen, “Markov processes in quantum lattice systems,” Repts. Math. Phys.,14, No. 2, 207–213 (1978).Google Scholar
  55. 55.
    M. D. Choi, “A Schwartz inequality for positive linear maps on C*-algebra,” Ill. J. Math.,18, No. 4, 565–574 (1974).Google Scholar
  56. 56.
    E. Christensen, “Generators of semigroups of completely positive maps,” Commun. Math. Phys.,62, No. 2, 167–171 (1978).Google Scholar
  57. 57.
    E. Christensen and D. E. Evans, “Cohomology of operator algebras and quantum dynamical semigroups,” J. London Math. Soc.,20, No. 2, 358–368 (1979).Google Scholar
  58. 58.
    J. P. Conze and N. Dang-Ngoc, “Ergodic theorems for noncommutative dynamical systems,” Invent. Math.,46, No. 1, 1–15 (1978).Google Scholar
  59. 59.
    E. B. Davies, “Quantum stochastic processes,” Commun. Math. Phys.,15, No. 4, 277–304 (1969).Google Scholar
  60. 60.
    E. B. Davies, “Quantum stochastic processes. II,” Commun. Math. Phys.,19, No. 2, 83–105 (1970).Google Scholar
  61. 61.
    E. B. Davies, “Quantum stochastic processes. III,” Commun. Math. Phys.,22, No. 1, 51–70 (1971).Google Scholar
  62. 62.
    E. B. Davies, “Diffusion for weakly coupled quantum oscillators,” Commun. Math. Phys.,27, No. 4, 309–325 (1972).Google Scholar
  63. 63.
    E. B. Davies, “Some contraction semigroups in quantum probability,” Z. Wahrscheinlichketistheorie Verw. Geb.,23, No. 4, 261–273 (1972).Google Scholar
  64. 64.
    E. B. Davies, “Exact dynamics of an infinite-atom Dicke maser model,” Commun. Math. Phys.,33, No. 3, 187–205 (1973).Google Scholar
  65. 65.
    E. B. Davies, “The infinite-atom Dicke maser model. II,” Commun. Math. Phys.,34, No. 3, 237–249 (1973).Google Scholar
  66. 66.
    E. B. Davies, “The harmonic oscillator in a heat bath,” Commun. Math. Phys.,33, No. 3, 171–186 (1973).Google Scholar
  67. 67.
    E. B. Davies, “Markovian master equations,” Commun. Math. Phys.,39, No. 2, 91–110 (1974).Google Scholar
  68. 68.
    E. B. Davies, “Markovian master equations. II,” Math. Ann.,219, No. 2, 147–158 (1976).Google Scholar
  69. 69.
    E. B. Davies, “Markovian master equations. III,” Ann. Inst. H. Poincaré,B11, No. 3, 265–273 (1975).Google Scholar
  70. 70.
    E. B. Davies, “The classical limit for quantum dynamical semigroups,” Commun. Math. Phys.,49, No. 2, 113–129 (1976).Google Scholar
  71. 71.
    E. B. Davies, “Quantum communication systems,” IEEE Trans. Inf. Theory,23, No. 4, 530–534 (1977).Google Scholar
  72. 72.
    E. B. Davies, “Irreversible dynamics of infinite fermion systems,” Commun. Math. Phys.,55, No. 3, 231–258 (1977).Google Scholar
  73. 73.
    E. B. Davies, “First and second quantized neutron diffusion equations,” Commun. Math. Phys.,52, No. 2, 111–126 (1977).Google Scholar
  74. 74.
    E. B. Davies, “Quantum dynamical semigroups and the neutron diffusion equation,” Repts. Math. Phys.,11, No. 2, 169–188 (1977).Google Scholar
  75. 75.
    E. B. Davies, Quantum Theory of Open Systems, Academic Press, London (1976).Google Scholar
  76. 76.
    E. B. Davies, “A model of heat conduction,” J. Statist. Phys.,18, No. 2, 161–170 (1978).Google Scholar
  77. 77.
    E. B. Davies, “Dilations of completely positive maps,” J. London Math. Soc.,17, No. 2, 330–338 (1978).Google Scholar
  78. 78.
    E. B. Davies, “Generators of dynamical semigroups,” J. Funct. Anal.,34, No. 3, 421–432 (1979).Google Scholar
  79. 79.
    E. B. Davies, “Uniqueness of the standard form of the generator of a quantum dynamical semigroup,” Repts. Math. Phys.,17, No. 2, 249–255 (1980).Google Scholar
  80. 80.
    E. B. Davies, “Asymptotic modifications of dynamical semigroups on C*-algebras,” J. London Math. Soc.,24, No. 3, 537–547 (1981).Google Scholar
  81. 81.
    E. B. Davies, One-Parameter Semigroups, Academic Press, London (1980).Google Scholar
  82. 82.
    E. B. Davies, and J. T. Lewis, “An operational approach to quantum probability,” Commun. Math. Phys.,17, No. 4, 239–260 (1970).Google Scholar
  83. 83.
    B. Demoen and P. Vanheuverzwijn, “Implenetable positive maps on standard forms,” J. Funct. Anal.,38, No. 3, 354–365 (1980).Google Scholar
  84. 84.
    B. Demoen, P. Vanheuverzwijn, and A. Verbeure, “Completely positive maps on the CCR-algebra,” Lett. Math. Phys.,2, No. 2, 161–166 (1977).Google Scholar
  85. 85.
    B. Demoen, P. Vanheuverzwijn, and A. Verbeure, “Energetically stable systems,” J. Math. Phys.,19, No. 11, 2256–2259 (1978).Google Scholar
  86. 86.
    R. Dumke and H. Spohn, “The proper form of the generator in the weak coupling limit,” Z. Phys.,34, No. 4, 419–422 (1982).Google Scholar
  87. 87.
    G. O. S. Ekhaguere, “Markov fields in noncommutative probability theory on W*-algebras,” J. Math. Phys.,20, No. 8, 1679–1683 (1979).Google Scholar
  88. 88.
    G. G. Emch, “The minimal K-flow associated to a quantum diffusion process,” in: Physical Reality and Mathematical Description, Dordrecht (1974), pp. 477–493.Google Scholar
  89. 89.
    G. G. Emch, “Positivity of the K-entropy of non-Abelian K-flows,” Z. Wahrscheinlichkeitstheor. Verw. Geb.,29, No. 3, 241–252 (1974).Google Scholar
  90. 90.
    G. G. Emch, “An algebraic approach to the theory of K-flows and K-entropy,” Lect. Notes Phys.,39, 315–318 (1975).Google Scholar
  91. 91.
    G. G. Emch, “Minimal dilations of CP-flows,” Lect. Notes Math.,650, 156–159 (1978).Google Scholar
  92. 92.
    G. G. Emch, “Some mathematical problems in nonequilibrium statistical mechanics,” Lect. Notes Math.,843, 264–295 (1981).Google Scholar
  93. 93.
    G. G. Emch, S. Albeverio, and J. P. Eckmann, “Quasifree generalized K-flows,” Repts. Math. Phys.,13, No. 1, 73–85 (1978).Google Scholar
  94. 94.
    G. G. Emch and J. Varilly, “On the standard form of the Bloch equation,” Lett. Math. Phys.,3, No. 2, 113–116 (1979).Google Scholar
  95. 95.
    M. Enomoto and Y. Watanani, “A Perron-Frobenius type theorem for positive linear maps on C*-algebras,” Math. Jpn.,24, No. 1, 53–63 (1979).Google Scholar
  96. 96.
    D. E. Evans, “Irreducible quantum dynamical semigroups,” Commun. Math. Phys.,54, No. 3, 293–297 (1977).Google Scholar
  97. 97.
    D. E. Evans, “Conditionally complete positive maps on operator algebras,” Q. J. Math.,28, No. 11, 271–284 (1977).Google Scholar
  98. 98.
    D. E. Evans, “A review on semigroups of completely positive maps,” Math. Probl. Theor. Phys. Proc. Int. Conf., Lausanne, 400–408 (1979).Google Scholar
  99. 99.
    D. E. Evans, “Completely positive quasi-free maps on the CAR algebra,” Commun. Math. Phys.,70, No. 1, 53–68 (1979).Google Scholar
  100. 100.
    D. E. Evans, “Dissipators for symmetric quasifree dynamical semigroups on the CAR algebra,” J. Funct. Anal.,37, No. 3, 318–330 (1980).Google Scholar
  101. 101.
    D. E. Evans and H. Hanche-Olsen, “The generators of positive semigroups,” J. Funct. Anal.,32, No. 2, 207–212 (1979).Google Scholar
  102. 102.
    D. E. Evans and R. Hoegh-Krohn, “Spectral properties of positive maps on C*-algebras,” J. London Math. Soc.,17, No. 2, 345–355 (1978).Google Scholar
  103. 103.
    D. E. Evans and J. T. Lewis, “Dilations of dynamical semigroups,” Commun. Math. Phys.,50, No. 3, 219–227 (1976).Google Scholar
  104. 104.
    D. E. Evans and J. T. Lewis, “Dilations of irreversible evolutions in algebraic quantum theory,” Commun. Dublin Inst. Adv. Stud., Ser. A, No. 24 (1977).Google Scholar
  105. 105.
    M. Fannes, R. Martens, and A. Verbeure, “Moments of the autocorrelation function and the KMS-condition,” Commun. Math. Phys.,80, No. 4, 529–541 (1981).Google Scholar
  106. 106.
    M. Fannes and F. Rocca, “A class of dissipative evolutions with applications in thermodynamics of fermion systems,” J. Math. Phys.,21, No. 2, 221–226 (1980).Google Scholar
  107. 107.
    A. Frigerio, “Quantum dynamical semigroups and approach to equilibrium,” Lett. Math. Phys.,2, No. 2, 79–87 (1977).Google Scholar
  108. 108.
    A. Frigerio, “Stationary states of quantum dynamical semigroups,” Commun. Math. Phys.,63, No. 3, 269–276 (1978).Google Scholar
  109. 109.
    A. Frigerio, V. Gorini, and J. V. Pulé, “Open quasifree systems,” J. Statist. Phys.,22, No. 4, 409–433 (1980).Google Scholar
  110. 110.
    A. Frigerio, C. Novellone, and M. Verri, “Master equation treatment of the singular reservoir limit,” Repts. Math. Phys.,12, No. 2, 279–284 (1977).Google Scholar
  111. 111.
    V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. C. G. Sudarshan, “Properties of quantum Markovian master equations,” Repts. Math. Phys.,13, No. 2, 149–173 (1978).Google Scholar
  112. 112.
    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of n-level systems,” J. Math. Phys.,17, No. 5, 821–825 (1976).Google Scholar
  113. 113.
    P. Grigolini, “Non-Markovian excitation-relaxation processes,” European Conference on the Dynamics of Excited States, Pisa, Apr. 14–16, 1980, Nuovo Cimento,B63, No. 1, 174–206 (1981).Google Scholar
  114. 114.
    U. Groh, “The periopheral point spectrum of Schwarz operators on C*-algebras,” Math. Z.,176, No. 3, 311–318 (1981).Google Scholar
  115. 115.
    C. Grosu, “The entropy of a quantum dynamical system,” An. Univ. Bucuresti Mat.,28, 37–49 (1979).Google Scholar
  116. 116.
    S. Gudder and J. P. Marchand, “Conditional expectations on von Neumann algebras: A new approach,” Repts. Math. Phys.,12, No. 3, 317–329 (1977).Google Scholar
  117. 117.
    W. Guz, “On quantum dynamical semigroups,” Repts. Math. Phys.,6, No. 3, 455–464 (1974).Google Scholar
  118. 118.
    W. Guz, “Markovian processes in classical and quantum statistical mechanics,” Repts. Math. Phys.,7, No. 2, 205–214 (1975).Google Scholar
  119. 119.
    W. Guz, “On time evolution of nonisolated physical systems,” Repts. Math. Phys.,8, No. 1, 49–59 (1975).Google Scholar
  120. 120.
    H. Hasegawa and T. Nakagomi, “On the characterization of the stationary state of a class of dynamical semigroups,” J. Statist. Phys.,23, No. 5, 639–652 (1980).Google Scholar
  121. 121.
    H. Hasegawa and T. Nakagomi, “Semiclassical laser theory in the stochastic and thermodynamic frameworks,” J. Statist. Phys.,21, No. 2, 191–214 (1979).Google Scholar
  122. 122.
    H. Hasegawa, T. Nakagomi, M. Mabuchi, and K. Rondo, “Nonequilibrium thermodynamics of lasing and bistable optical systems,” J. Statist. Phys.,23, No. 3, 281–313 (1980).Google Scholar
  123. 123.
    K. Hepp and E. H. Lieb, “The laser: a reversible quantum dynamical system with irreversible classical macroscopic motion,” Lect. Notes Phys.,38, 178–207 (1975).Google Scholar
  124. 124.
    A. S. Holevo, “Problems in the mathematical theory of quantum communications channels,” Repts. Math. Phys.,12, No. 2, 273–278 (1977).Google Scholar
  125. 125.
    R. S. Ingarden and A. Kassakowski, “On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems,” Ann. Phys. (USA),89, No. 2, 451–485 (1975).Google Scholar
  126. 126.
    K. Jezuita and B. S. Skagerstam, “Nonequilibrium quantum statistical mechanics of a damped harmonic oscillator,” Repts. Math. Phys.,15, No. 3, 423–431 (1979).Google Scholar
  127. 127.
    P. E. T. Jorgensen, “The existence problem for dynamics in the C*-algebraic formulation of dissipative quantum systems,” Preprint Ser. Mat. Inst. Aarhus Univ., No. 1, (1980–1981).Google Scholar
  128. 128.
    A. Kossakowski, “On quantum statistical mechanics of non-Hamiltonian systems,” Repts. Path. Phys.,3, No. 4, 247–274 (1972).Google Scholar
  129. 129.
    A. Kossakowski, A. Frigerio, V. Gorini, and M. Verri, “Quantum detailed balance KMS condition,” Commun. Math. Phys.,60, No. 1, 96 (1978).Google Scholar
  130. 130.
    K. Kraus, “General state changes in quantum theory,” Ann. Phys.,64, No. 2, 311–333 (1971).Google Scholar
  131. 131.
    B. Kümmerer, “A noncommutative individual ergodic theorem,” Invent. Math.,46, No. 2, 139–145 (1978).Google Scholar
  132. 132.
    G. Lindblad, “An entropy inequality for quantum measurements,” Commun. Math. Phys.,28, No. 3, 245–249 (1972).Google Scholar
  133. 133.
    G. Lindblad, “Expectations and entropy inequalities for finite quantum systems,” Commun. Math. Phys.,39, No. 2, 111–119 (1974).Google Scholar
  134. 134.
    G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys.,48, No. 2, 119–130 (1976).Google Scholar
  135. 135.
    G. Lindblad, “Brownian motion of a quantum harmonic oscillator,” Repts. Math. Phys.,10, No. 3, 219–224 (1976).Google Scholar
  136. 136.
    G. Lindblad, “Dissipative operators and cohomology of operator algebras,” Lett. Math. Phys.,1, No. 3, 219–224 (1976).Google Scholar
  137. 137.
    G. Lindblad, “Non-Markovian quantum stochastic processes and their entropy,” Commun. Math. Phys.,65, No. 3, 281–294 (1979).Google Scholar
  138. 138.
    G. Lindblad, “Gaussian quantum stochastic processes on the CCR algebra,” J. Math. Phys.,20, No. 10, 2081–2087 (1979).Google Scholar
  139. 139.
    S. Mukamel, I. Openheim, and J. Ross, “Statistical reduction for strongly driven simple quantum systems,” Phys. Rev.,A17, No. 6, 1988–1998 (1978).Google Scholar
  140. 140.
    H. Narnhofer, “Scattering theory for quasifree time automorphisms of C*-algebras and von Neumann algebras,” Repts. Math. Phys.,16, No. 1, 1–9 (1979).Google Scholar
  141. 141.
    J. Naudts, J. V. Pulé, and A. Verbeure, “Linear response and relaxation in quantum lattice systems,” J. Statist. Phys.,21, No. 3, 279–288 (1979).Google Scholar
  142. 142.
    M. Ohya, “On linear-response dynamics,” Lett. Nuovo Cimento,21, No. 16, 573–576 (1978).Google Scholar
  143. 143.
    M. Ohya, “Dynamical process in linear response theory,” Repts. Math. Phys.,16, No. 3, 305–315 (1979).Google Scholar
  144. 144.
    M. Ohya, “An open system dynamics. An operator algebraic study,” Kodai Math. J.,3, No. 2, 287–294 (1980).Google Scholar
  145. 145.
    P. F. Palmer, “The singular coupling and the weak coupling limits,” J. Math. Phys.,18, No. 3, 527–529 (1977).Google Scholar
  146. 146.
    P. F. Palmer, “Characterizations of *-homomorphisms and expectations,” Proc. Am. Math. Soc.,46, No. 2, 265–272 (1974).Google Scholar
  147. 147.
    G. C. Papanicolaou and W. Kohler, “Asymptotic analysis of determinatic and stochastic equations and rapidly varying components,” Commun. Math. Phys.,45, No. 3, 217–232 (1975).Google Scholar
  148. 148.
    G. Parravicin and A. Zecca, “On the generator of completely positive dynamical semigroups of N-level systems,” Repts. Math. Phys.,12, No. 3, 423–424 (1977).Google Scholar
  149. 149.
    G. Piron, “Survey of general quantum physics,” Log.-Algebraic Approach Quantum Mech., Vol. 1, Histor. Evol., Dordrecht-Boston (1975), pp. 513–543.Google Scholar
  150. 150.
    A. Posiewnik, “On quantum Markovian processes,” Repts. Math. Phys.,12, No. 2, 229–236 (1977).Google Scholar
  151. 151.
    A. Posiewnik, “Extension on quantum dynamical semigroups,” Repts. Math. Phys.,15, No. 3, 417–420 (1979).Google Scholar
  152. 152.
    A. Posiewnik, “One property of dynamical semigroups,” Repts. Math. Phys.,15, No. 3, 421–422 (1979).Google Scholar
  153. 153.
    E. Presutti, F. Scacciatelli, G. L. Sewell, and F. Wanderlingh, “Studies in the C*-algebraic theory of nonequilibrium statistical mechanics: dynamics of open and of mechanically driven systems,” J. Math. Phys.,13, No. 8, 1085–1098 (1972).Google Scholar
  154. 154.
    J. V. Pulé, “The Bloch equations,” Commun. Math. Phys.,38, No. 3, 241–256 (1974).Google Scholar
  155. 155.
    J. V. Pulé, “Some properties of the correlation function of quantum stochastic processes,” Proc. Cambr. Phil. Soc.,76, No. 3, 607–612 (1974).Google Scholar
  156. 156.
    J. V. Pulé and A. Verbeure, “The classical limit of quantum dissipative generators,” J. Math. Phys.,20, No. 4, 733–735 (1979).Google Scholar
  157. 157.
    J. V. Pulé and A. Verbeure, “Dissipative operators for infinite classical systems and equilibrium,” J. Math. Phys.,20, No. 11, 2286–2290 (1979).Google Scholar
  158. 158.
    R. R. Puri and S. V. Lawande, “On exact master equation for an open system.,” Phys. Lett.,A62, No. 3, 143–145 (1977).Google Scholar
  159. 159.
    W. Pusz and S. L. Woronowicz, “Passive states and KMS states for general quantum systems,” Commun. Math. Phys.,58, No. 3, 273–290 (1978).Google Scholar
  160. 160.
    W. Pusz and S. L. Woronowicz, “Form convex functions and the Wydl and other inequalities,” Lett. Math. Phys.,2, No. 6, 505–512 (1978).Google Scholar
  161. 161.
    H. Roos, “On quantum systems in thermal contact,” Commun. Math. Phys.,26, No. 2, 149–168 (1972).Google Scholar
  162. 162.
    R. Schrader and D. A. Uhlenbrock, “Markov structures on Clifford algebras” J. Funct Anal.,18, No. 4, 369–413 (1975).Google Scholar
  163. 163.
    H. Scutaru, “Some remarks on covariant completely positive linear maps on C*-algebras. Repts. Math. Phys.,16, No. 1, 79–87 (1979).Google Scholar
  164. 164.
    H. Spohn, “Approach to equilibrium for completely positive dynamical semigroups of N-level systems,” Repts. Math. Phys.,10, No. 2, 189–194 (1976).Google Scholar
  165. 165.
    H. Spohn, “An algebraic condition for the approach to equilibrium of an open N-level system,” Lett. Math. Phys.,2, No. 1, 33–38 (1977).Google Scholar
  166. 166.
    H. Spohn, “Entropy production for quantum dynamical semi groups” J. Math. Phy.19 No. 5, 1227–1230 (1978).Google Scholar
  167. 167.
    M. D. Srinivas, “Foundations of a quantum probability theory,” J. Math. Phys,18, No. 8, 1672–1685 (1975).Google Scholar
  168. 168.
    M. D. Srinivas, “Quantum counting processes” J. Math. Phys.,18, No. 11, 2138–2145 (1977).Google Scholar
  169. 169.
    M. D. Srinivas, “Conditional probabilities and-statistical independence in quantum theory,” J. Math. Phys.,19, No. 8, 1705–1710 (1978).Google Scholar
  170. 170.
    M. D. Srinivas, “Collapse postulate for observables with continuous spectra” Commun. Math. Phys.,71, No. 2, 131–158 (1980).Google Scholar
  171. 171.
    W. F. Stinesprings “Positive functions on C*-algebras,” Proc. Am. Math. Soc.,6, No. 2, 211–216 (1955).Google Scholar
  172. 172.
    E. Stormer, “Positive linear maps of C*-algebras,” Lect. Notes Phys.,29, 85–106 (1974).Google Scholar
  173. 173.
    E. C. G. Sudarshan, T. N. Sherry, and S. R. Gautam, “An approach to measurement in quantum mechanics,” Particles and Fields. Proc. Banff Summer Inst., Banff, 1977, New York-London (1978), pp. 289–304.Google Scholar
  174. 174.
    P. Talkner, “Untersuchungen irreversibler Prozesse in quantenmechanischen Systemen” Diss. Dokt. Naturwiss. Univ. Stuttgart (1979).Google Scholar
  175. 175.
    C. W. Thompson, “Dissipations on von Neumann algebras,” Commun. Math. Phys.,52, No. 1, 71–78 (1978).Google Scholar
  176. 176.
    V. D. Vainstein and S. D. Tvorogov, “Some problems on the measurement of quantum observables and determination of joint entropy in quantum statistics” Commun. Math. Phys.,43, No. 3, 273–278 (1975).Google Scholar
  177. 177.
    J. C. Varilly, “Dilation on a nonquasi-free dissipative evolution,” Lett. Math. Phys.,5, No. 2, 113–116 (1981).Google Scholar
  178. 178.
    A. Verbeure, “Characterization of equilibrium states.,” Colloq. Int. CNRS, No. 274, 441–450 (1979).Google Scholar
  179. 179.
    M. Verri and V. Gorini, “Quantum dynamical semigroups and multipole relaxation of a spin in isotropic surroundings,” J. Math. Phys.,19, No., 9, 1803–1807 (1978).Google Scholar
  180. 180.
    Y. Watatani, “Kernels, potentials and complete positivity,” Math. Jpn.,24, No., 1, 93–96 (1979).Google Scholar
  181. 181.
    A. Wehrl, “On the relation between classical and quantum-mechanical entropy,” Repts. Math. Phys.,16, No. 3, 353–358 (1979).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • V. I. Oseledets

There are no affiliations available

Personalised recommendations