Asymptotic dynamics in the Heisenberg picture: attractor subspace and Choi-Effros product

TL;DR

This work analyzes the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture, providing an explicit structure for the attractor subspace $\mathrm{Attr}(\Phi)$ and the long-time evolution on it. By decomposing the Hilbert space into a faithful recurrent part and a transient part, the authors show how the Choi-Effros product endows $\mathrm{Attr}(\Phi)$ with a ${C}^*$-algebra structure and reveal an unfolding theorem that links Heisenberg asymptotics to a reduced faithful map. The paper further derives a decomposition for the Choi-Effros decoherence-free algebra $\mathcal{N}_{\star}$, and extends the analysis to Schwarz maps, illustrating that operator Schwarz inequality suffices to capture the essential asymptotic structure. Altogether, the results bridge Schrödinger and Heisenberg pictures, clarify the algebraic underpinnings of long-time open-system dynamics, and broaden applicability to broader map classes relevant for decoherence-free subspaces and reservoir engineering.

Abstract

We study the asymptotic dynamics of open quantum systems in the Heisenberg picture. We find an explicit expression for the attractor subspace and the dynamics that takes place in it. We present the relationship between the attractor subspaces in the Schrödinger and Heisenberg pictures and, in particular, the connection between their algebraic structures. An unfolding theorem of the asymptotics, as well as the fine structure of the recently introduced Choi-Effros decoherence-free algebra, are also discussed. Finally, we show how to extend all the results to the class of Schwarz maps.