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arXiv:2511.17770 · Open Systems & Information Dynamics

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

Daniele Amato, Paolo Facchi, Arturo Konderak

quant-phmath-phSubmitted Nov 21, 2025• Announced Nov 25, 2025• 1 citation
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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.

Journal reference
Open System & Information Dynamics Vol.32, No. 03, 2550014, 2025
DOI
10.1142/S1230161225500143
Comments
22 pages

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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.
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The dissipative dynamics anticipated in the proof of 't Hooft's existence theorem — "For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization" — is constructed here explicitly. We propose a generalization of Liouville's classical phase space equation, incorporating dissipation and diffusion, and demonstrate that it describes the emergence of quantum states and their dynamics in the Schrödinger picture. Asymptotically, there is a stable ground state and two decoupled sets of degrees of freedom, which transform into each other under the energy-parity symmetry of Kaplan and Sundrum. They recover the familiar Hilbert space and its dual. Expectations of observables are shown to agree with the Born rule, which is not imposed a priori. This attractor mechanism is applicable in the presence of interactions, to few-body or field theories in particular.
Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrödinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.
In this note, we develop a framework to describe open quantum systems in the Heisenberg picture, i.e., via time evolving operator algebras. We point out the incompleteness of the previous proposals in this regard. We argue that a complete Heisenberg picture for an open quantum system involves multiple image Heisenberg operators for each system observable. For a given system observable, the number of such image operators is equal to the dimension of the environment Hilbert space. We derive a perturbative expression, accurate upto arbitrary orders in the system environment coupling, for these image operators in terms of a single one point operator. This expression depends non-linearly on the state of the environment. This perturbative expression can equivalently be thought of as deforming the operator product on the Hilbert space of the open quantum system. In the Markovian limit, the one point operator evolves by an adjoint Lindblad equation. We illustrate these ideas using a simple spin system.
In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.