The cosmic background gravitational wave radiation does not set a specific upper limit for preparing a coherent system based on its size, such as fullerene molecules. Coherence refers to the quantum mechanical property of a system, indicating the wave-like behavior and the ability to exist in a superposition of states. It is not directly dependent on gravitational wave radiation or its effects.
Gravitational waves are ripples in the fabric of spacetime caused by accelerating massive objects. While they can interact with matter and energy, their influence on the coherence of quantum systems is typically negligible. Gravitational waves primarily affect large-scale systems or extremely massive objects, such as black holes or neutron stars. The effects of gravitational waves on individual molecules or smaller systems are extremely weak and difficult to detect.
Therefore, the size of the system or the preparation of a coherent system is not fundamentally limited by cosmic background gravitational wave radiation. The factors that primarily determine the coherence of a quantum system are related to its isolation, control of external interactions, and preservation of quantum states, such as minimizing environmental noise, temperature, and decoherence processes.