How Far do Lindbladians Go?

TL;DR

The paper investigates how little dissipation is needed to make the full state space $\\mathcal{D}(\\mathcal{H})$ reachable when unitary control is already powerful (Hörmander generation of $\\mathfrak{su}(n)$). It develops a geometric and algebraic framework around Lindbladians, tangent cones, and lifting to characterize reachability, proving that a small number of non-self-adjoint jump operators suffices for transitivity in many cases, while purely self-adjoint jumps fail. It further incorporates environment-assisted resources and a gradient-form perspective to expand the set of attainable evolutions, and it provides constructive algorithms for state transport using sparse Lindbladians and amplitude damping, including finite-time reachability results and obstructions. The results reframe dissipation as a controllable resource for universal state preparation and reservoir engineering, with practical implications for designing efficient open-system control protocols.

Abstract

We study controllability of finite-dimensional open quantum systems under a general Markovian control model combining full coherent (unitary) control with tunable dissipative channels. Assuming the Hamiltonian controls is a Hörmander system that generate $\mathfrak{su}(n)$, we ask how little dissipation suffices to make the full state space $\mathcal{D}(\mathcal{H})$ controllable. We show that minimal non-unital noise can break unitary-orbit invariants and, in many cases, a very small set of jump operators yields transitivity on $\mathcal{D}(\mathcal{H})$. For multi-qubit systems we prove explicit transitivity results for natural resources such as a single-qubit amplitude-damping jump together with a dephasing channel, and we identify obstructions when only self-adjoint jump operators are available (yielding only unital evolutions). We further develop a geometric viewpoint and ask the ``lifting'' question: when can a path of densities be obtained from applying a time-dependent family of Lindbladian to an initial state? For this, we have to analyze the tangent structure of the ``manifold with corners'' and how this tangent structure reflects Lindbldian evolution. Building on this framework, we derive reachability criteria and no-go results based on a norm-decrease alignment condition, including a geometric obstruction arising from the incompatibility between admissible tangent directions and dissipative contraction.

Figures (2)

• Figure 1: Illustration of the alignment condition of \ref{['p2condition']}
• Figure 2: A choice of raising Lindbladians on the commutative state space of $\mathbb C^3$.

Theorems & Definitions (61)

• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Proposition 2.7