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

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 ${\displaystyle {\mathcal {A}}}$ be a von Neumann algebra acting on Hilbert space ${\displaystyle {\mathcal {H}}}$, a quantum dynamical semigroup on ${\displaystyle {\mathcal {A}}}$ is a collection of bounded operators on ${\displaystyle {\mathcal {A}}}$, denoted by ${\displaystyle {\mathcal {T}}:=\left({\mathcal {T}}_{t}\right)_{t\geq 0}}$, with the following properties: ^[5]

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

Under the condition of complete positivity, the operators ${\displaystyle {\mathcal {T}}_{t}}$ are ${\displaystyle \sigma }$-weakly continuous if and only if ${\displaystyle {\mathcal {T}}_{t}}$ are normal. ^[5] Recall that, letting ${\displaystyle {\mathcal {A}}_{+}}$ denote the convex cone of positive elements in ${\displaystyle {\mathcal {A}}}$, a positive operator ${\displaystyle T:{\mathcal {A}}\rightarrow {\mathcal {A}}}$ is said to be normal if for every increasing net ${\displaystyle \left(x_{\alpha }\right)_{\alpha }}$ in ${\displaystyle {\mathcal {A}}_{+}}$ with least upper bound ${\displaystyle x}$ in ${\displaystyle {\mathcal {A}}_{+}}$ one has

${\displaystyle \lim _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\sup _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\langle u,(Tx)u\rangle }$

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

Quantum Markov semigroup (QMS)

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

${\displaystyle {\mathcal {T}}_{t}\left({\boldsymbol {1}}\right)={\boldsymbol {1}},\quad \forall t\geq 0,}$

1

where ${\displaystyle {\boldsymbol {1}}\in {\mathcal {A}}}$ is the identity element. For simplicity, ${\displaystyle {\mathcal {T}}}$ is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of ${\displaystyle {\mathcal {T}}_{t}}$ imply ${\displaystyle \left\|{\mathcal {T}}_{t}\right\|=1}$ for all ${\displaystyle t\geq 0}$ and then ${\displaystyle {\mathcal {T}}}$ 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 Hudson–Parthasarathy 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 ${\displaystyle {\mathcal {T}}}$ is the operator ${\displaystyle {\mathcal {L}}}$ with domain ${\displaystyle \operatorname {Dom} ({\mathcal {L}})}$, where

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

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

Characterization of generators of uniformly continuous QMSs

Main article: Lindbladian

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

• the infinitesimal generator ${\displaystyle {\mathcal {L}}}$ will be a bounded operator on von Neumann algebra ${\displaystyle {\mathcal {A}}}$ with domain ${\displaystyle \mathrm {Dom} ({\mathcal {L}})={\mathcal {A}}}$, ^[8]
• the map ${\displaystyle t\mapsto {\mathcal {T}}_{t}a}$ will automatically be continuous for every ${\displaystyle a\in {\mathcal {A}}}$, ^[8]
• the infinitesimal generator ${\displaystyle {\mathcal {L}}}$ will be also ${\displaystyle \sigma }$-weakly continuous. ^[9]

Under such assumption, the infinitesimal generator ${\displaystyle {\mathcal {L}}}$ has the characterization ^[3]

${\displaystyle {\mathcal {L}}\left(a\right)=i\left[H,a\right]+\sum _{j}\left(V_{j}^{\dagger }aV_{j}-{\frac {1}{2}}\left\{V_{j}^{\dagger }V_{j},a\right\}\right)}$

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

See also

• 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 functionPages displaying short descriptions of redirect targets
• Contraction semigroup  – Generalization of the exponential functionPages displaying short descriptions of redirect targets
• 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

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. 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. 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. 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).