Symmetry & Controllability for Spin Networks with a Single-Node Control

TL;DR

This paper investigates how symmetries affect the controllability of spin networks with XXZ coupling when controlling a single node, revealing conditions for full controllability and the influence of network structure.

Contribution

It provides a rigorous analysis of symmetries and controllability in spin networks with single-node Z-control, including characterizations for uniform chains and branched networks.

Findings

External symmetries are characterized by eigenstates with zero overlap at the control node.

Symmetries can persist despite random perturbations.

Lack of symmetry is both necessary and sufficient for subspace controllability in uniform chains.

Abstract

We consider the relation of symmetries and subspace controllability for spin networks with XXZ coupling subject to control of a single node by a local potential (Z-control). Such networks decompose into excitation subspaces. Focusing on the single excitation subspace it is shown that for single-node Z-controls external symmetries are characterized by eigenstates of the system Hamiltonian that have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries that persist even in the presence of random perturbations. For uniformly coupled XXZ chains a characterization of all possible symmetries is given, which shows a strong dependence on the position of the node we control. Finally, it is shown rigorously for uniform Heisenberg and XX chains subject to single-node Z-control that the lack of symmetry is not only necessary but sufficient for subspace controllability. The latter approach is then generalized to establish controllability results for simple branched networks.