We show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals. This holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left half-plane, whilst for eigenvalues in the open right half-plane, we present a counterexample in which the strong solution to the associated algebraic Riccati equation is not diffusive. For nonidentical systems satisfying the internal model principle for synchronization, we show that a certain part of the coupling must be diffusive. For equally chosen weights in the cost functional, we show that the dimension of the associated algebraic Riccati equation can be reduced significantly.