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Journal of Mathematical Physics

The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups

Franco Fagnola, Emanuela Sasso, Veronica Umanità

Published Jul 1, 2019• 13 citations
Publisher

Abstract

We show that for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we prove that the predual of the decoherence-free subalgebra is isometrically isomorphic to the subspace of reversible states. We also describe, in an explicit and constructive way, the relationship between the decoherence-free subalgebra and the fixed point subalgebra.

DOI
10.1063/1.5030954

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We show that for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we prove that the predual of the decoherence-free subalgebra is isometrically isomorphic to the subspace of reversible states. We also describe, in an explicit and constructive way, the relationship between the decoherence-free subalgebra and the fixed point subalgebra.
We establish the structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra and apply the result to derive a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. We deduce the structure of invariant states and a method for finding decoherence-free subsystems and subspaces. The relationship between environment induced decoherence and the decoherence-free subalgebra is also discussed.
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.