Decoherence-free algebras in quantum dynamics

TL;DR

This paper explores the algebraic structure of the long-term behavior of finite-dimensional open quantum systems, introducing the Choi-Effros decoherence-free algebra and analyzing its properties and relation to attractor subspaces.

Contribution

It introduces the Choi-Effros decoherence-free algebra, a novel algebraic structure capturing asymptotic quantum dynamics, independent of complete positivity assumptions.

Findings

The Choi-Effros algebra is both a C*-algebra and a B*-algebra.

The algebra admits a direct-sum decomposition linked to the attractor subspace.

Equality of attractor subspace and the Choi-Effros algebra characterizes faithful dynamics.

Abstract

In this Article we analyze the algebraic properties of the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. In particular, a natural product (Choi-Effros product) can be defined in the asymptotic regime. Motivated by this structure, we introduce a new space called the Choi-Effros decoherence-free algebra. Interestingly, this space is both a C* -algebra with respect to the composition product, and a B* -algebra with respect to the Choi-Effros product. Moreover, such space admits a direct-sum decomposition revealing a clear relationship with the attractor subspace of the dynamics. In particular, the equality between the attractor subspace and the Choi-Effros decoherence-free algebra is a necessary and sufficient condition for a faithful dynamics. Finally, we show how all the findings do not rely on complete positivity but on the much weaker Schwarz property.