Reachability, Coolability, and Stabilizability of Open Markovian Quantum Systems with Fast Unitary Control

TL;DR

The paper develops a rigorous framework linking open Markovian quantum dynamics with fast unitary control to a reduced control system on the eigenvalue simplex $\Delta^{n-1}$. By analyzing GKSL generators via relaxation algebras, inducing simplex dynamics $\dot\lambda=-L_U\lambda$, and proving an Equivalence Theorem between full and reduced models, it provides complete characterizations of stabilizability, viability, accessibility, and reachability, including asymptotic coolability and conditions under which time-independent Hamiltonians suffice. Special attention is given to unital systems, where the relaxation algebra yields strong structural results, enabling precise criteria for stabilizability, reachability, and controllability in terms of block structure and majorization. The findings have direct implications for cooling, state preparation, and controllability in quantum technologies, highlighting when simple, time-invariant controls can achieve robust objectives despite dissipation. Overall, the work offers a principled bridge between abstract Lindblad-algebra properties and practical control tasks on open quantum systems.

Abstract

Open Markovian quantum systems with fast and full Hamiltonian control can be reduced to an equivalent control system on the standard simplex modelling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate applications to the cooling of Markovian quantum systems. We show that for certain tasks of interest, the control Hamiltonian can be chosen time-independent. -- The reduction picture is an example of dissipative interconversion between equivalence classes of states, where the classes are induced by fast controls.

Figures (1)

• Figure 1: Relationship between the time evolutions of a bilinear control system \ref{['eq:bilinear-control-system']} on density matrices $\rho(t)$ and the reduced control system \ref{['eq:simplex-control-system']} governing the dynamics of the eigenvalues of $\rho(t)$, where "the" vector $\lambda(t)$ of eigenvalues is depicted by the respective diagonal matrix ${\rm diag}(\lambda(t))$, see Section \ref{['sec:tools']}. The derivative $\dot\rho(t)$ can always be split into a part orthogonal to the orbit (using the orthogonal projection $\Pi_{\rho(t)}$ onto the commutant of $\rho(t)$), and a part tangent to the orbit (using the complementary projection $\Pi_{\rho(t)}^\perp$). We depict only the regular case, where $\Pi_{\rho(t)}={\rm Ad}_{U(t)}\circ\Pi_{{\rm diag}}\circ{\rm Ad}_{U(t)}^{-1}$ for $\rho(t)=U(t){\rm diag}(\lambda(t))U(t)^*$. A central result in this work is the Equivalence Theorem \ref{['thm:equivalence']} in the main text which details the equivalence of the two control systems.

Theorems & Definitions (114)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4: Equivalence Theorem
• Corollary 2.5
• Proof 1
• Proposition 2.6
• Proof 2
• Definition 3.1
• Proposition 3.2