View source for Quantum Markov semigroup

{{Short description|A kind of mathematical structure which describes the dynamics in a Markovian open quantum system.}}

In [[quantum mechanics]], a '''quantum Markov semigroup''' describes the dynamics in a [[Markov Property|Markovian]] [[open quantum system]]. The axiomatic definition of the prototype of '''quantum Markov semigroups''' was first introduced by [[Andrzej Kossakowski|A. M. Kossakowski]]<ref name="Kossakowski">{{cite journal |last1=Kossakowski |first1=A. |title=On quantum statistical mechanics of non-Hamiltonian systems |journal=Reports on Mathematical Physics |date=December 1972 |volume=3 |issue=4 |pages=247–274 |doi=10.1016/0034-4877(72)90010-9|bibcode=1972RpMP....3..247K }}</ref> in 1972, and then developed by V. Gorini, [[Andrzej Kossakowski|A. M. Kossakowski]], [[E. C. George Sudarshan|E. C. G. Sudarshan]]<ref name="GKS1976">{{cite journal |last1=Gorini |first1=Vittorio |last2=Kossakowski |first2=Andrzej |last3=Sudarshan |first3=Ennackal Chandy George |title=Completely positive dynamical semigroups of N-level systems |journal=Journal of Mathematical Physics |date=1976 |volume=17 |issue=5 |pages=821 |doi=10.1063/1.522979|bibcode=1976JMP....17..821G }}</ref> and [[Göran Lindblad (physicist)|Göran Lindblad]]<ref name="Lindbladian" /> in 1976.<ref name="History">{{cite journal |last1=Chruściński |first1=Dariusz |last2=Pascazio |first2=Saverio |title=A Brief History of the GKLS Equation |journal=Open Systems & Information Dynamics |date=September 2017 |volume=24 |issue=3 |pages=1740001 |doi=10.1142/S1230161217400017|arxiv=1710.05993 |bibcode=2017OSID...2440001C |s2cid=90357 }}</ref>

==Motivation==
An ideal [[quantum system]] is not realistic because it should be completely isolated while, in practice, it is influenced by the [[quantum coupling|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 operator|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 [[Lindbladian|Lindblad equation]] or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few [[quantum noise]]s. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of [[one-parameter group]]s 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 algebra]]s, so the dimensionality of the system could be infinite. Let <math> \mathcal{A} </math> be a von Neumann algebra acting on [[Hilbert space]] <math> \mathcal{H} </math>, a quantum dynamical semigroup on <math> \mathcal{A} </math> is a collection of bounded operators on <math> \mathcal{A} </math>, denoted by <math> \mathcal{T} := \left( \mathcal{T}_t \right)_{t \ge 0} </math>, with the following properties:<ref name="QMS-FF">{{cite journal |last1=Fagnola |first1=Franco |title=Quantum Markov semigroups and quantum flows |journal=Proyecciones |date=1999 |volume=18 |issue=3 |pages=1–144 |doi=10.22199/S07160917.1999.0003.00002 |url=https://www.researchgate.net/publication/247317142|doi-access=free }}</ref> 
# <math> \mathcal{T}_0 \left( a \right) = a </math>, <math> \forall a \in \mathcal{A} </math>,
# <math> \mathcal{T}_{t + s} \left( a \right) = \mathcal{T}_t \left( \mathcal{T}_s \left( a \right) \right) </math>, <math> \forall s, t \ge 0 </math>, <math> \forall a \in \mathcal{A} </math>,
# <math> \mathcal{T}_t </math> is [[completely positive map|completely positive]] for all <math> t \ge 0 </math>,
# <math> \mathcal{T}_t </math> is a <math> \sigma </math>-weakly continuous operator in <math> \mathcal{A} </math> for all <math> t \ge 0 </math>,
# For all <math> a \in \mathcal{A} </math>, the map <math> t \mapsto \mathcal{T}_t \left( a \right) </math> is continuous with respect to the <math> \sigma </math>-weak topology on <math> \mathcal{A} </math>.
Under the condition of complete positivity, the operators <math> \mathcal{T}_t </math> are <math> \sigma </math>-weakly continuous if and only if <math> \mathcal{T}_t </math> are normal.<ref name="QMS-FF" /> Recall that, letting <math> \mathcal{A}_+ </math> denote the [[convex cone]] of positive elements in <math> \mathcal{A} </math>, a positive operator <math> T : \mathcal{A} \rightarrow \mathcal{A} </math> is said to be normal if for every increasing [[net (mathematics)|net]] <math> \left( x_\alpha \right)_\alpha </math>  in <math> \mathcal{A}_+ </math> with [[least upper bound]] <math> x </math> in <math> \mathcal{A}_+ </math> one has 
:<math> \lim_{\alpha} \langle u, (T x_\alpha) u \rangle = \sup_{\alpha} \langle u, (T x_\alpha) u \rangle = \langle u, (T x) u \rangle </math>
for each <math> u </math> in a [[dense set|norm-dense]] [[linear manifold|linear sub-manifold]] of <math> \mathcal{H} </math>.

===Quantum Markov semigroup (QMS)===

A quantum dynamical semigroup <math> \mathcal{T} </math> is said to be identity-preserving (or conservative, or Markovian) if
{{NumBlk|:|<math> \mathcal{T}_t \left( \boldsymbol{1} \right) = \boldsymbol{1}, \quad \forall t \ge 0, </math>|{{EquationRef|1}}}}
where <math> \boldsymbol{1} \in \mathcal{A} </math> is the identity element. For simplicity, <math> \mathcal{T} </math> is called quantum Markov semigroup. Notice that, the identity-preserving property and [[positive element|positivity]] of <math> \mathcal{T}_t </math> imply <math> \left\| \mathcal{T}_t \right\| = 1 </math> for all <math> t \ge 0 </math> and then <math> \mathcal{T} </math> is a [[contraction semigroup]].<ref name="Operator-alg-Bratteli">{{cite book |last1=Bratteli |first1=Ola |last2=Robinson |first2=Derek William |title=Operator algebras and quantum statistical mechanics |date=1987 |publisher=Springer-Verlag |location=New York |isbn=3-540-17093-6 |edition=2nd}}</ref>

The Condition ({{EquationNote|1}}) plays an important role not only in the proof of uniqueness and unitarity of solution of a [[Robin Lyth Hudson|Hudson]]–[[K. R. Parthasarathy (probabilist)|Parthasarathy]] [[quantum stochastic calculus|quantum stochastic differential equation]], but also in deducing regularity conditions for paths of classical Markov processes in view of [[operator theory]].<ref name="MinimalQDS-AC-FF">{{cite journal |last1=Chebotarev |first1=A.M |last2=Fagnola |first2=F |title=Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups |journal=Journal of Functional Analysis |date=March 1998 |volume=153 |issue=2 |pages=382–404 |doi=10.1006/jfan.1997.3189|arxiv=funct-an/9711006 |s2cid=18823390 }}</ref>

===Infinitesimal generator of QDS===
The infinitesimal generator of a quantum dynamical semigroup <math> \mathcal{T} </math> is the operator <math> \mathcal{L} </math> with domain <math> \operatorname{Dom} (\mathcal{L}) </math>, where
:<math> \operatorname{Dom} \left( \mathcal{L} \right) := \left\{ a \in \mathcal{A} ~\left\vert~ \lim_{t \rightarrow 0} \frac{\mathcal{T}_t(a) - a}{t} = b \text{ in } \sigma\text{-weak topology} \right. \right\} </math>
and <math> \mathcal{L}(a) := b </math>.

==Characterization of generators of uniformly continuous QMSs==

{{Main|Lindbladian}}

If the quantum Markov semigroup <math> \mathcal{T} </math> is uniformly continuous in addition, which means <math> \lim_{t \rightarrow 0^+} \left\| \mathcal{T}_t - \mathcal{T}_0 \right\| = 0 </math>, then 
* the infinitesimal generator <math> \mathcal{L} </math> will be a [[bounded operator]] on von Neumann algebra <math> \mathcal{A} </math> with domain <math> \mathrm{Dom} (\mathcal{L}) = \mathcal{A} </math>,<ref name="FA-Rudin">{{cite book |last1=Rudin |first1=Walter |title=Functional analysis |date=1991 |publisher=McGraw-Hill Science/Engineering/Math |location=New York |isbn=978-0070542365 |edition=Second}}</ref>
* the map <math> t \mapsto \mathcal{T}_t a </math> will automatically be continuous for every <math> a \in \mathcal{A} </math>,<ref name="FA-Rudin" />
* the infinitesimal generator <math> \mathcal{L} </math> will be also <math> \sigma </math>-weakly continuous.<ref name="Diximier-sigma-weak-continuity">{{cite journal |last1=Dixmier |first1=Jacques |title=Les algèbres d'opérateurs dans l'espace hilbertien |journal=Mathematical Reviews (MathSciNet) |date=1957}}</ref>

Under such assumption, the infinitesimal generator <math> \mathcal{L} </math> has the characterization<ref name="Lindbladian">{{cite journal |last1=Lindblad |first1=Goran |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |date=1976 |volume=48 |issue=2 |pages=119–130 |doi=10.1007/BF01608499|bibcode=1976CMaPh..48..119L |s2cid=55220796 |url=http://projecteuclid.org/euclid.cmp/1103899849 }}</ref>
:<math> \mathcal{L} \left( a \right) = i \left[ H, a \right] + \sum_{j} \left( V_j^\dagger a V_j - \frac{1}{2} \left\{ V_j^\dagger V_j, a \right\} \right) </math>
where <math> a \in \mathcal{A} </math>, <math> V_j \in \mathcal{B} (\mathcal{H}) </math>, <math> \sum_{j} V_j^\dagger V_j \in \mathcal{B} (\mathcal{H}) </math>, and <math> H \in \mathcal{B} (\mathcal{H}) </math> is [[self-adjoint operator|self-adjoint]]. Moreover, above <math> \left[ \cdot, \cdot \right] </math> denotes the [[commutator]], and <math> \left\{ \cdot, \cdot \right\} </math> the [[commutator|anti-commutator]].

==Selected recent publications==

* {{cite journal |last1=Chebotarev |first1=A.M |last2=Fagnola |first2=F |title=Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups |journal=Journal of Functional Analysis |date=March 1998 |volume=153 |issue=2 |pages=382–404 |doi=10.1006/jfan.1997.3189|arxiv=funct-an/9711006 |s2cid=18823390 }}
* {{cite journal |last1=Fagnola |first1=Franco |last2=Rebolledo |first2=Rolando |title=Transience and recurrence of quantum Markov semigroups |journal=Probability Theory and Related Fields |date=2003-06-01 |volume=126 |issue=2 |pages=289–306 |doi=10.1007/s00440-003-0268-0|s2cid=123052568 |doi-access=free }}
* {{cite journal |last1=Rebolledo |first1=R |title=Decoherence of quantum Markov semigroups |journal=Annales de l'Institut Henri Poincaré B |date=May 2005 |volume=41 |issue=3 |pages=349–373 |doi=10.1016/j.anihpb.2004.12.003|bibcode=2005AIHPB..41..349R |url=http://www.numdam.org/item/AIHPB_2005__41_3_349_0/ }}
* {{cite journal |last1=Umanità |first1=Veronica |title=Classification and decomposition of Quantum Markov Semigroups |journal=Probability Theory and Related Fields |date=April 2006 |volume=134 |issue=4 |pages=603–623 |doi=10.1007/s00440-005-0450-7|s2cid=119409078 |doi-access=free }}
* {{cite journal |last1=Fagnola |first1=Franco |last2=Umanità |first2=Veronica |title=Generators of detailed balance quantum markov semigroups |journal=Infinite Dimensional Analysis, Quantum Probability and Related Topics |date=2007-09-01 |volume=10 |issue=3 |pages=335–363 |doi=10.1142/S0219025707002762|arxiv=0707.2147 |s2cid=16690012 }}
* {{cite journal |last1=Carlen |first1=Eric A. |last2=Maas |first2=Jan |title=Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance |journal=Journal of Functional Analysis |date=September 2017 |volume=273 |issue=5 |pages=1810–1869 |doi=10.1016/j.jfa.2017.05.003|arxiv=1609.01254 |s2cid=119734534 }}

==See also==

* {{Annotated link|Operator topologies}}
* {{Annotated link|Von Neumann algebra}}
* {{Annotated link|C0 semigroup}}
* {{Annotated link|Contraction semigroup}}
* {{Annotated link|Lindbladian}}
* {{Annotated link|Markov chain}}
* {{Annotated link|Quantum mechanics}}
* {{Annotated link|Open quantum system}}

==References==
{{reflist}}

[[Category:Quantum mechanics]]
[[Category:Semigroup theory]]