RECONSTRUCTION THEOREM FOR STATIONARY MONOTONE QUANTUM MARKOV PROCESSES

Title & Authors
RECONSTRUCTION THEOREM FOR STATIONARY MONOTONE QUANTUM MARKOV PROCESSES
Heo, Jae-Seong; Belavkin, Viacheslav P.; Ji, Un Cig;

Abstract
Based on the Hilbert $\small{C^*}$-module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.
Keywords
Hilbert $\small{C^*}$-module;covariant representation;quantum stochastic process;quantum dynamical semigroup;
Language
English
Cited by
1.
Stochastic Processes and Spectral Analysis for Hilbert \$\$C^*\$\$ C ∗ -Module-Valued Maps, Bulletin of the Malaysian Mathematical Sciences Society, 2015
References
1.
L. Accardi, A. Frigerio, and J. T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 97-133.

2.
V. P. Belavkin, A reconstruction theorem for a quantum random field, Uspekhi Mat. Nauk 39 (1984), no. 2, 137-138.

3.
V. P. Belavkin, Reconstruction theorem for a quantum stochastic process, Theor. Math. Phys. 62 (1985), 275-289.

4.
B. V. Bhat and K. R. Parthasarathy, Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory, Ann. Inst. H. Poincare Probab. Statist. 31(1995), no. 4, 601-651.

5.
B. V. Bhat and M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 4, 519-575.

6.
P. S. Chakraborty, D. Goswami, and K. B. Sinha, A covariant quantum stochastic dilation theory, Stochastics in finite and infinite dimensions, 89-99, Trends Math., Birkhauser, Boston, 2001.

7.
E. Christensen and D. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. Soc. (2) 20 (1979), no. 2, 358-368.

8.
E. B. Davies, Quantum stochastic processes, Comm. Math. Phys. 15 (1969), 277-304.

9.
E. B. Davies, Markovian master equations, Comm. Math. Phys. 39 (1974), 91-110.

10.
D. Goswami and K. B. Sinha, Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra, Comm. Math. Phys. 205 (1999), no. 2, 377-403.

11.
J. Heo, Completely multi-positive linear maps and representations on Hilbert \$C^{\ast}\$- modules, J. Operator Theory 41 (1999), no. 1, 3-22.

12.
J. Heo, Hilbert \$C^{\ast}\$-module representation on Haagerup tensor products and group systems, Publ. Res. Inst. Math. Sci. 35 (1999), no. 5, 757-768.

13.
J. Heo, V. P. Belavkin, and U. C. Ji, Monotone quantum stochastic processes and covariant dynamical hemigroups, J. Func. Anal. 261 (2011), 3345-3365.

14.
E. Lance, Hilbert \$C^{\ast}\$-modules, Cambridge University Press, 1995.

15.
G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), no. 2, 119-130.

16.
G. Lindblad, Non-Markovian quantum stochastic processes and their entropy, Comm. Math. Phys. 65 (1979), no. 3, 281-294.

17.
K. Parthasarathy, A continuous time version of Stinespring's theorem on completely positive maps, Quantum probability and applications, V (Heidelberg, 1988), 296-300, Lecture Notes in Math., 1442, Springer, Berlin, 1990.

18.
W. Paschke, Inner product modules over \$B^{\ast}\$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468.