Particle-number threshold for non-Abelian geometric phases

When a quantum state traverses a path, while being under the influence of a gauge potential, it acquires a geometric phase that is often more than just a scalar quantity. The variety of unitary transformations that can be realized by this form of parallel transport depends crucially on the number of particles involved in the evolution. Here, we introduce a particle-number threshold (PNT) that assesses a system's capabilities to perform purely geometric manipulations of quantum states. This threshold gives the minimal number of particles necessary to fully exploit a system's potential to generate non-Abelian geometric phases. Therefore, the PNT might be useful for evaluating the resource demands of a holonomic quantum computer. We benchmark our findings on bosonic systems relevant to linear and nonlinear quantum optics.