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Zbl 1209.46041
Dhahri, Ameur; Fagnola, Franco; Rebolledo, Rolando
The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator. (English)
[J] Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13, No. 3, 413-433 (2010). ISSN 0219-0257

Summary: Let $\cal T$ be a quantum Markov semigroup on ${\cal B}(h)$ with a faithful normal invariant state $\rho$. The decoherence-free subalgebra ${\cal N}'({\cal T})$ of $\cal T$ is the biggest subalgebra of ${\cal B}(h)$ where the completely positive maps ${\cal T}_t$ act as homomorphisms. When $\cal T$ is the minimal semigroup whose generator is represented in a generalised GKSL form ${\cal L}(x) =-\frac12\sum_\ell(L^*_\ell L_\ell x-2L^*_\ell xL_\ell+xL^*_\ell L_\ell)+i[H,x]$ with possibly unbounded $H$ and $L_\ell$, we show that ${\cal N}({\cal L})$ coincides with the generalised commutator of $\{e^{-itH}L_\ell e^{-itH} L_\ell e^{itH}\mid\ell\ge 1$, $t\ge 0\}$ under some natural regularity conditions. As a corollary, we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of $H$ and $L_\ell$. We give examples of quantum Markov semigroups ${\cal B}({\frak h})$, with $\frak h$ infinite-dimensional, having a nontrivial decoherence-free subalgebra.

MSC 2000:: *46L55 Noncommutative dynamical systems
46L57 Derivations etc. in $C^*$-algebras
47D07 Markov semigroups of linear operators
47N50 Appl. of operator theory in quantum physics
81S25 Quantum stochastic calculus
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)

Keywords: quantum Markov semigroups; decoherence; convergence to a steady state; generalised Lindblad form; multiple commutators

Cited in: Zbl 1243.46054

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