Figure 2: Schematics of the full versus projected system algebras.
From: Exponential rise of dynamical complexity in quantum computing through projections
The arrows are tangents (generators) on a manifold of unitary transformations. In the larger space (upper), the operations commute, so no matter which way we go, we end up at the same point. It is not the case for the projected system (lower): the projected operations do not commute, and the gap represents the non-commutativity. Even though the projected system is embedded in a smaller space, its dynamics is more complex, because of the curvature induced by the projection: new directions can be explored.
