# Subspace controllability of bipartite symmetric spin networks under global control

## TL;DR

This paper analyzes the controllability of bipartite symmetric spin networks with global control, demonstrating subspace controllability in systems relevant to NV centers in diamonds.

## Contribution

It characterizes invariant subspaces and the dynamical Lie algebra, proving subspace controllability for these spin networks with symmetric interactions.

## Key findings

- Invariant subspaces are explicitly characterized.
- Dynamical Lie algebra is determined for the network.
- Subspace controllability is established for all cases.

## Abstract

We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.