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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. Kossakowski in 1972, and then developed by V. 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. Kossakowski in 1972, and then developed by V. 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).

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 $𝒜$ be a von Neumann algebra acting on Hilbert space $ℋ$ , a quantum dynamical semigroup on $𝒜$ is a collection of bounded operators on $𝒜$ , denoted by $𝒯 : = {(𝒯_{t})}_{t \geq 0}$ , with the following properties:^[1]

$𝒯_{0} (a) = a$ , $\forall a \in 𝒜$ ,
$𝒯_{t + s} (a) = 𝒯_{t} (𝒯_{s} (a))$ , $\forall s, t \geq 0$ , $\forall a \in 𝒜$ ,
$𝒯_{t}$ is completely positive for all $t \geq 0$ ,
$𝒯_{t}$ is a $σ$ -weakly continuous operator in $𝒜$ for all $t \geq 0$ ,
For all $a \in 𝒜$ , the map $t \mapsto 𝒯_{t} (a)$ is continuous with respect to the $σ$ -weak topology on $𝒜$ .

It is worth mentioning that, under the condition of complete positivity, the operators $𝒯_{t}$ are $σ$ -weakly continuous if and only if $𝒯_{t}$ are normal.^[1] Recall that, letting $𝒜_{+}$ denote the convex cone of positive elements in $𝒜$ , a positive operator $T : 𝒜 \to 𝒜$ is said to be normal if for every increasing net ${(x_{α})}_{α}$ in $𝒜_{+}$ with least upper bound $x$ in $𝒜_{+}$ one has

\lim_{α} ⟨ u, (T x_{α}) u ⟩ = \sup_{α} ⟨ u, (T x_{α}) u ⟩ = ⟨ u, (T x) u ⟩

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

Quantum Markov semigroup (QMS)

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

$𝒯_{t} (1) = 1, \forall t \geq 0,$

(1)

where $1 \in 𝒜$ is the identity element. For simplicity, $𝒯$ is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of $𝒯_{t}$ imply $‖ 𝒯_{t} ‖ = 1$ for all $t \geq 0$ and then $𝒯$ is a contraction semigroup.^[2]

The Condition (1) plays an important role not only in the proof of uniqueness and unitarity of solution of a Hudson–Parthasarathy quantum stochastic differential equation, but also in deducing regularity conditions for paths of classical Markov processes in view of operator theory.^[3]

Infinitesimal generator of QDS

The infinitesimal generator of a quantum dynamical semigroup $𝒯$ is the operator $ℒ$ with domain $Dom (ℒ)$ , where

Dom (ℒ) : = {a \in 𝒜 | \lim_{t \to 0} \frac{𝒯_{t} (a) - a}{t} = b in σ -weak topology}

and $ℒ (a) : = b$ .

Characterization of generators of uniformly continuous QMSs

Related topic: Lindbladian

If the quantum Markov semigroup $𝒯$ is uniformly continuous in addition, which means $\lim_{t \to 0^{+}} ‖ 𝒯_{t} - 𝒯_{0} ‖ = 0$ , then

the infinitesimal generator $ℒ$ will be a bounded operator on von Neumann algebra $𝒜$ with domain $D o m (ℒ) = 𝒜$ ,^[4]
the map $t \mapsto 𝒯_{t} a$ will automatically be continuous for every $a \in 𝒜$ ,^[4]
the infinitesimal generator $ℒ$ will be also $σ$ -weakly continuous.^[5]

Under such assumption, the infinitesimal generator $ℒ$ has the characterization^[6]

ℒ (a) = i [H, a] + \sum_{j} (V_{j}^{†} a V_{j} - \frac{1}{2} {V_{j}^{†} V_{j}, a})

where $a \in 𝒜$ , $V_{j} \in ℬ (ℋ)$ , $\sum_{j} V_{j}^{†} V_{j} \in ℬ (ℋ)$ , and $H \in ℬ (ℋ)$ is self-adjoint. Moreover, above $[\cdot, \cdot]$ denotes the commutator, and ${\cdot, \cdot}$ the anti-commutator.

@@ Line 1: / Line 1: @@
 {{Short description|A kind of mathematical structure which describes the dynamics in a Markovian open quantum system.}}
 {{Quantum book backlink|Open quantum systems}}
+{{Quantum article nav|previous=Physics:Quantum dissipation|previous label=Dissipation|next=Physics:Quantum Markovian dynamics|next label=Markovian dynamics}}
 <div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
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 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-In [[Physics:Quantum mechanics|quantum mechanics]], a '''quantum Markov semigroup''' describes the dynamics in a Markovian [[Physics:Open quantum system|open quantum system]]. The axiomatic definition of the prototype of '''quantum Markov semigroups''' was first introduced by 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}}</ref> in 1972, and then developed by V. Gorini, A. M. Kossakowski, 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}}</ref> and [[Biography: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 |s2cid=90357 }}</ref>
+'''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. Kossakowski in 1972, and then developed by V. 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. Kossakowski in 1972, and then developed by V. 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).
 </div>
 <div style="width:300px;">
-[[File:not found]]
+[[File:Quantum Markov semigroup.png|thumb|240px]]
 </div>
@@ Line 20: / Line 19: @@
 ==Motivation==
-An ideal [[Physics:Quantum system|quantum system]] is not realistic because it should be completely isolated while, in practice, it is influenced by the [[Physics:Quantum coupling|coupling]] to an environment, which typically has a large number of degrees of freedom (for example an [[Physics:Atom|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 [[Physics:Schrödinger equation|Schrödinger equation]] is replaced by a suitable [[Physics:Master equation|master equation]], such as a [[Physics:Lindbladian|Lindblad equation]] or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few [[Physics:Quantum noise|quantum noise]]s. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of [[One-parameter group|one-parameter group]]s of unitary maps, but one needs to introduce '''quantum Markov semigroups'''.
+An ideal [[Physics:Quantum system|quantum system]] is not realistic because it should be completely isolated while, in practice, it is influenced by the [[Physics: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 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 [[Physics:Quantum noise|quantum noise]]s. 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==
@@ Line 26: / Line 25: @@
 ===Quantum dynamical semigroup (QDS)===
-In general, quantum dynamical semigroups can be defined on [[Von Neumann algebra|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>
+In general, quantum dynamical semigroups can be defined on von Neumann algebras, 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>,
@@ Line 32: / Line 31: @@
 # <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>.
-It is worth mentioning that, 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|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
+It is worth mentioning that, 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 <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 sub-manifold of <math> \mathcal{H} </math>.
+for each <math> u </math> in a norm-dense linear sub-manifold of <math> \mathcal{H} </math>.
 ===Quantum Markov semigroup (QMS)===
@@ Line 40: / Line 39: @@
 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>
+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 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 [[Biography:Robin Lyth Hudson|Hudson]]–[[Biography: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|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>
+The Condition ({{EquationNote|1}}) plays an important role not only in the proof of uniqueness and unitarity of solution of a Hudson–Parthasarathy 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===
@@ Line 51: / Line 50: @@
 ==Characterization of generators of uniformly continuous QMSs==
-{{Main|Physics:Lindbladian}}
+''Related topic:'' 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|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 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>
@@ Line 60: / Line 59: @@
 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|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|commutator]], and <math> \left\{ \cdot, \cdot \right\} </math> the [[Commutator|anti-commutator]].
+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. Moreover, above <math> \left[ \cdot, \cdot \right] </math> denotes the commutator, and <math> \left\{ \cdot, \cdot \right\} </math> the anti-commutator.
 ==Selected recent publications==
@@ Line 71: / Line 70: @@
 * {{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==
+== See also ==
+{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/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==
@@ Line 88: / Line 79: @@
 [[Category:Semigroup theory]]
-{{Sourceattribution|Quantum Markov semigroup}}
+{{Author|Harold Foppele}}
+{{Sourceattribution|Physics:Quantum Markov semigroup|1}}

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Physics:Quantum Markov semigroup: Difference between revisions

Latest revision as of 22:20, 23 May 2026

Motivation

Definitions

Quantum dynamical semigroup (QDS)

Quantum Markov semigroup (QMS)

Infinitesimal generator of QDS

Characterization of generators of uniformly continuous QMSs

Selected recent publications

See also

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