Can entanglement be mediated by a Koopmanian system?

We present a method for coupling a Koopmanian classical system to two quantum bits to mediate an interaction between them. We then prove that the resulting dynamics can never lead to entanglement between the two qubits. Even though the total system of two qubits and the Koopmanian classical system are described with the full quantum formalism, we show that their composite system violates exact conservation laws as expected for a hybrid quantum-classical system. We finally discuss the implications for semi-classical treatments of quantum gravity.