Quantum Markov semigroup

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.

In general, quantum dynamical semigroups can be defined on von Neumann algebras, so the dimensionality of the system could be infinite. Let $\backslash mathcal\{A\}$ be a von Neumann algebra acting on Hilbert space $\backslash mathcal\{H\}$, a quantum dynamical semigroup on $\backslash mathcal\{A\}$ is a collection of bounded operators on $\backslash mathcal\{A\}$, denoted by $\backslash mathcal\{T\}\; :=\; \backslash left(\; \backslash mathcal\{T\}\_t\; \backslash right)\_\{t\; \backslash ge\; 0\}$, with the following properties:{{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 }}

1. $\backslash mathcal\{T\}\_0\; \backslash left(\; a\; \backslash right)\; =\; a$, $\backslash forall\; a\; \backslash in\; \backslash mathcal\{A\}$,
2. $\backslash mathcal\{T\}\_\{t\; +\; s\}\; \backslash left(\; a\; \backslash right)\; =\; \backslash mathcal\{T\}\_t\; \backslash left(\; \backslash mathcal\{T\}\_s\; \backslash left(\; a\; \backslash right)\; \backslash right)$, $\backslash forall\; s,\; t\; \backslash ge\; 0$, $\backslash forall\; a\; \backslash in\; \backslash mathcal\{A\}$,
3. $\backslash mathcal\{T\}\_t$ is completely positive for all $t\; \backslash ge\; 0$,
4. $\backslash mathcal\{T\}\_t$ is a $\backslash sigma$-weakly continuous operator in $\backslash mathcal\{A\}$ for all $t\; \backslash ge\; 0$,
5. For all $a\; \backslash in\; \backslash mathcal\{A\}$, the map $t\; \backslash mapsto\; \backslash mathcal\{T\}\_t\; \backslash left(\; a\; \backslash right)$ is continuous with respect to the $\backslash sigma$-weak topology on $\backslash mathcal\{A\}$.

Under the condition of complete positivity, the operators $\backslash mathcal\{T\}\_t$ are $\backslash sigma$-weakly continuous if and only if $\backslash mathcal\{T\}\_t$ are normal. Recall that, letting $\backslash mathcal\{A\}\_+$ denote the convex cone of positive elements in $\backslash mathcal\{A\}$, a positive operator $T\; :\; \backslash mathcal\{A\}\; \backslash rightarrow\; \backslash mathcal\{A\}$ is said to be normal if for every increasing net $\backslash left(\; x\_\backslash alpha\; \backslash right)\_\backslash alpha$ in $\backslash mathcal\{A\}\_+$ with least upper bound $x$ in $\backslash mathcal\{A\}\_+$ one has

:$\backslash lim\_\{\backslash alpha\}\; \backslash langle\; u,\; (T\; x\_\backslash alpha)\; u\; \backslash rangle\; =\; \backslash sup\_\{\backslash alpha\}\; \backslash langle\; u,\; (T\; x\_\backslash alpha)\; u\; \backslash rangle\; =\; \backslash langle\; u,\; (T\; x)\; u\; \backslash rangle$

for each $u$ in a norm-dense linear sub-manifold of $\backslash mathcal\{H\}$.

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

{{NumBlk|:|$\backslash mathcal\{T\}\_t\; \backslash left(\; \backslash boldsymbol\{1\}\; \backslash right)\; =\; \backslash boldsymbol\{1\},\; \backslash quad\; \backslash forall\; t\; \backslash ge\; 0,$|{{EquationRef|1}}}}

where $\backslash boldsymbol\{1\}\; \backslash in\; \backslash mathcal\{A\}$ is the identity element. For simplicity, $\backslash mathcal\{T\}$ is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of $\backslash mathcal\{T\}\_t$ imply $\backslash left\backslash |\; \backslash mathcal\{T\}\_t\; \backslash right\backslash |\; =\; 1$ for all $t\; \backslash ge\; 0$ and then $\backslash mathcal\{T\}$ is a contraction semigroup.{{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}}

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.{{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 }}

The infinitesimal generator of a quantum dynamical semigroup $\backslash mathcal\{T\}$ is the operator $\backslash mathcal\{L\}$ with domain $\backslash operatorname\{Dom\}\; (\backslash mathcal\{L\})$, where

:$\backslash operatorname\{Dom\}\; \backslash left(\; \backslash mathcal\{L\}\; \backslash right)\; :=\; \backslash left\backslash \{\; a\; \backslash in\; \backslash mathcal\{A\}\; ~\backslash left\backslash vert~\; \backslash lim\_\{t\; \backslash rightarrow\; 0\}\; \backslash frac\{\backslash mathcal\{T\}\_t(a)\; -\; a\}\{t\}\; =\; b\; \backslash text\{\; in\; \}\; \backslash sigma\backslash text\{-weak\; topology\}\; \backslash right.\; \backslash right\backslash \}$

and $\backslash mathcal\{L\}(a)\; :=\; b$.

{{Main|Lindbladian}}

If the quantum Markov semigroup $\backslash mathcal\{T\}$ is uniformly continuous in addition, which means $\backslash lim\_\{t\; \backslash rightarrow\; 0^+\}\; \backslash left\backslash |\; \backslash mathcal\{T\}\_t\; -\; \backslash mathcal\{T\}\_0\; \backslash right\backslash |\; =\; 0$, then

• the infinitesimal generator $\backslash mathcal\{L\}$ will be a bounded operator on von Neumann algebra $\backslash mathcal\{A\}$ with domain $\backslash mathrm\{Dom\}\; (\backslash mathcal\{L\})\; =\; \backslash mathcal\{A\}$,{{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}}
• the map $t\; \backslash mapsto\; \backslash mathcal\{T\}\_t\; a$ will automatically be continuous for every $a\; \backslash in\; \backslash mathcal\{A\}$,
• the infinitesimal generator $\backslash mathcal\{L\}$ will be also $\backslash sigma$-weakly continuous.{{cite journal |last1=Dixmier |first1=Jacques |title=Les algèbres d'opérateurs dans l'espace hilbertien |journal=Mathematical Reviews (MathSciNet) |date=1957}}

Under such assumption, the infinitesimal generator $\backslash mathcal\{L\}$ has the characterization{{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 }}

:$\backslash mathcal\{L\}\; \backslash left(\; a\; \backslash right)\; =\; i\; \backslash left[\; H,\; a\; \backslash right]\; +\; \backslash sum\_\{j\}\; \backslash left(\; V\_j^\backslash dagger\; a\; V\_j\; -\; \backslash frac\{1\}\{2\}\; \backslash left\backslash \{\; V\_j^\backslash dagger\; V\_j,\; a\; \backslash right\backslash \}\; \backslash right)$

where $a\; \backslash in\; \backslash mathcal\{A\}$, $V\_j\; \backslash in\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\})$, $\backslash sum\_\{j\}\; V\_j^\backslash dagger\; V\_j\; \backslash in\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\})$, and $H\; \backslash in\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\})$ is self-adjoint. Moreover, above $\backslash left[\; \backslash cdot,\; \backslash cdot\; \backslash right]$ denotes the commutator, and $\backslash left\backslash \{\; \backslash cdot,\; \backslash cdot\; \backslash right\backslash \}$ the anti-commutator.

• {{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/ |hdl=10533/176263 |hdl-access=free }}
• {{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 }}

• {{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}}

{{reflist}}

Category:Quantum mechanics

Category:Semigroup theory