text: If we measure the spins separately, we get either \(\ket{\uparrow\uparrow}\) or one of the three other options, and all these states are non-entangled! The non-entanglement is guaranteed already when we decide that we measure these "isolated properties" of the two subsystems. On the other hand, if we measure the total spin or something like that whose eigenstates are entangled states of the two spins, we're guaranteed to bring the wave function into an entangled state after the measurement.