Quantum Markov semigroup

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

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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

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Quantum dynamical semigroup (QDS)

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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  .

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)

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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

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

 

and  .

Characterization of generators of uniformly continuous QMSs

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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

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  • 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. Bibcode:2005AIHPB..41..349R. 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

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  • Operator topologies – Topologies on the set of operators on a Hilbert space
  • Von Neumann algebra – *-algebra of bounded operators on a Hilbert space
  • C0 semigroup – Generalization of the exponential function
  • Contraction semigroup – Generalization of the exponential function
  • Lindbladian – Markovian quantum master equation for density matrices (mixed states)
  • Markov chain – Random process independent of past history
  • Quantum mechanics – Description of physical properties at the atomic and subatomic scale
  • Open quantum system – Quantum mechanical system that interacts with a quantum-mechanical environment

References

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  1. ^ Kossakowski, A. (December 1972). "On quantum statistical mechanics of non-Hamiltonian systems". Reports on Mathematical Physics. 3 (4): 247–274. Bibcode:1972RpMP....3..247K. 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. Bibcode:1976JMP....17..821G. doi:10.1063/1.522979.
  3. ^ a b Lindblad, Goran (1976). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics. 48 (2): 119–130. Bibcode:1976CMaPh..48..119L. 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. Bibcode:2017OSID...2440001C. doi:10.1142/S1230161217400017. S2CID 90357.
  5. ^ a b 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. ^ a b 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).