In quantum mechanics, a quantum Markov semigroup describes the dynamics in a Markovian open quantum system. The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski[1] in 1972, and then developed by V. Gorini, A. M. Kossakowski, E. C. G. Sudarshan[2] and Göran Lindblad[3] in 1976.[4]

Motivation

An ideal quantum system is not realistic because it should be completely isolated while, in practice, it is influenced by the coupling to an environment, which typically has a large number of degrees of freedom (for example an atom interacting with the surrounding radiation field). A complete microscopic description of the degrees of freedom of the environment is typically too complicated. Hence, one looks for simpler descriptions of the dynamics of the open system. In principle, one should investigate the unitary dynamics of the total system, i.e. the system and the environment, to obtain information about the reduced system of interest by averaging the appropriate observables over the degrees of freedom of the environment. To model the dissipative effects due to the interaction with the environment, the Schrödinger equation is replaced by a suitable master equation, such as a Lindblad equation or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few quantum noises. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of one-parameter groups of unitary maps, but one needs to introduce quantum Markov semigroups.

Definitions

Quantum dynamical semigroup (QDS)

In general, quantum dynamical semigroups can be defined on von Neumann algebras, so the dimensionality of the system could be infinite. Let be a von Neumann algebra acting on Hilbert space , a quantum dynamical semigroup on is a collection of bounded operators on , denoted by , with the following properties:[5]

  1. , ,
  2. , , ,
  3. is completely positive for all ,
  4. is a -weakly continuous operator in for all ,
  5. For all , the map is continuous with respect to the -weak topology on .

It is worth mentioning that, under the condition of complete positivity, the operators are -weakly continuous if and only if are normal.[5] Recall that, letting denote the convex cone of positive elements in , a positive operator is said to be normal if for every increasing net in with least upper bound in one has

for each in a norm-dense linear sub-manifold of .

Quantum Markov semigroup (QMS)

A quantum dynamical semigroup is said to be identity-preserving (or conservative, or Markovian) if

 

 

 

 

(1)

where is the identity element. For simplicity, is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of imply for all and then is a contraction semigroup.[6]

The Condition (1) plays an important role not only in the proof of uniqueness and unitarity of solution of a HudsonParthasarathy quantum stochastic differential equation, but also in deducing regularity conditions for paths of classical Markov processes in view of operator theory.[7]

Infinitesimal generator of QDS

The infinitesimal generator of a quantum dynamical semigroup is the operator with domain , where

and .

Characterization of generators of uniformly continuous QMSs

If the quantum Markov semigroup is uniformly continuous in addition, which means , then

  • the infinitesimal generator will be a bounded operator on von Neumann algebra with domain ,[8]
  • the map will automatically be continuous for every ,[8]
  • the infinitesimal generator will be also -weakly continuous.[9]

Under such assumption, the infinitesimal generator has the characterization[3]

where , , , and is self-adjoint. Moreover, above denotes the commutator, and the anti-commutator.

Selected recent publications

  • Chebotarev, A.M; Fagnola, F (March 1998). "Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups". Journal of Functional Analysis. 153 (2): 382–404. arXiv:funct-an/9711006. doi:10.1006/jfan.1997.3189. S2CID 18823390.
  • Fagnola, Franco; Rebolledo, Rolando (2003-06-01). "Transience and recurrence of quantum Markov semigroups". Probability Theory and Related Fields. 126 (2): 289–306. doi:10.1007/s00440-003-0268-0. S2CID 123052568.
  • Rebolledo, R (May 2005). "Decoherence of quantum Markov semigroups". Annales de l'Institut Henri Poincaré B. 41 (3): 349–373. doi:10.1016/j.anihpb.2004.12.003.
  • Umanità, Veronica (April 2006). "Classification and decomposition of Quantum Markov Semigroups". Probability Theory and Related Fields. 134 (4): 603–623. doi:10.1007/s00440-005-0450-7. S2CID 119409078.
  • Fagnola, Franco; Umanità, Veronica (2007-09-01). "Generators of detailed balance quantum markov semigroups". Infinite Dimensional Analysis, Quantum Probability and Related Topics. 10 (3): 335–363. arXiv:0707.2147. doi:10.1142/S0219025707002762. S2CID 16690012.
  • Carlen, Eric A.; Maas, Jan (September 2017). "Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance". Journal of Functional Analysis. 273 (5): 1810–1869. arXiv:1609.01254. doi:10.1016/j.jfa.2017.05.003. S2CID 119734534.

See also

References

  1. Kossakowski, A. (December 1972). "On quantum statistical mechanics of non-Hamiltonian systems". Reports on Mathematical Physics. 3 (4): 247–274. doi:10.1016/0034-4877(72)90010-9.
  2. Gorini, Vittorio; Kossakowski, Andrzej; Sudarshan, Ennackal Chandy George (1976). "Completely positive dynamical semigroups of N-level systems". Journal of Mathematical Physics. 17 (5): 821. doi:10.1063/1.522979.
  3. 1 2 Lindblad, Goran (1976). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics. 48 (2): 119–130. doi:10.1007/BF01608499. S2CID 55220796.
  4. Chruściński, Dariusz; Pascazio, Saverio (September 2017). "A Brief History of the GKLS Equation". Open Systems & Information Dynamics. 24 (3): 1740001. arXiv:1710.05993. doi:10.1142/S1230161217400017. S2CID 90357.
  5. 1 2 Fagnola, Franco (1999). "Quantum Markov semigroups and quantum flows". Proyecciones. 18 (3): 1–144. doi:10.22199/S07160917.1999.0003.00002.
  6. Bratteli, Ola; Robinson, Derek William (1987). Operator algebras and quantum statistical mechanics (2nd ed.). New York: Springer-Verlag. ISBN 3-540-17093-6.
  7. Chebotarev, A.M; Fagnola, F (March 1998). "Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups". Journal of Functional Analysis. 153 (2): 382–404. arXiv:funct-an/9711006. doi:10.1006/jfan.1997.3189. S2CID 18823390.
  8. 1 2 Rudin, Walter (1991). Functional analysis (Second ed.). New York: McGraw-Hill Science/Engineering/Math. ISBN 978-0070542365.
  9. Dixmier, Jacques (1957). "Les algèbres d'opérateurs dans l'espace hilbertien". Mathematical Reviews (MathSciNet).
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