Attractor Subspace and Decoherence-Free Algebra of Quantum Dynamics
Daniele Amato, Paolo Facchi, Arturo Konderak
TL;DR
This work analyzes the long-time behavior of open quantum systems in the Heisenberg picture, bridging spectral and algebraic descriptions of decoherence through two complementary viewpoints: the peripheral-eigenstructure of quantum channels and the decoherence-free subalgebra on which dynamics act as ∗-automorphisms. It shows that, for faithful Markovian evolutions, the attractor subspace and the decoherence-free algebra coincide, ensuring the equivalence of the spectral and algebraic approaches, with unitary evolution on the decoherence-free sector. The GKLS framework clarifies how the asymptotics separate into a reversible (unitary) decoherence-free part and a transient dissipative part, governed by the peripheral spectrum and fixed points. In infinite dimensions, new phenomena emerge, including the possibility that the decoherence-free algebra is a non-atomic von Neumann algebra, such as type II_1 or type III factors, highlighting richer asymptotics and connections to quantum field theory. Overall, the article provides a coherent bridge between mathematical structure and physical intuition for decoherence-free dynamics and their implications for quantum information and field-theoretic contexts.
Abstract
In this review we discuss some results on the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. Both the spectral and algebraic approaches to this topic are addressed, with particular emphasis on their relationship. The analysis is conducted in both the discrete-time and the continuous-time Markovian settings. In the final part of the work, some issues emerging in the infinite-dimensional case are also discussed. In particular, we provide an example of a Markovian evolution whose decoherence-free algebra is a type III von Neumann algebra.