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THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP ON $\mathcal{B}({\rm{h}})$

Ameur Dhahri, Franco Fagnola, Rolando Rebolledo

Published Jan 1, 2011• 0 citations
Publisher
DOI
10.1142/9789814338745_0007

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THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

Ameur Dhahri, Franco Fagnola, Rolando Rebolledo

Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] 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 Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Sep 1, 2010•45 citations
THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP ON B(h)\mathcal{B}({\rm{h}})B(h)
Generators of KMS Symmetric Markov Semigroups on B(h){\mathcal{B}({\rm h})}B(h) Symmetry and Quantum Detailed Balance
This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.
Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] 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 Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.