Consider a system composed of two spin-1/2 particles. The total spin operator is defined as $$ \mathbf{S} = \mathbf{S_1} + \mathbf{S_2} $$ We can write a common eigenbasis of the operators $\mathbf{S^2}, S_z, \mathbf{S_1^2}$ and $ \mathbf{S_2^2}$ in terms of the eigenbasis of the operators $S_{1z}, S_{2z}, \mathbf{S_1^2}$ and $ \mathbf{S_2^2}$, which we denote by $|\pm\pm\rangle$ This gives a spin triplet a a spin singlet: $$ |s = 1, m = 1\rangle = |++\rangle\\ |s = 1, m = 0\rangle = \frac{1}{\sqrt{2}}(|+-\rangle + |- + \rangle)\\ |s = 1, m = 1\rangle = |--\rangle\\ |s = 0, m = 0\rangle = \frac{1}{\sqrt{2}}(|+-\rangle - |- + \rangle) $$ Where the following eigenvalue relations hold: $$ \mathbf{S^2}|s, m\rangle = s(s + 1)\hbar^2|s, m\rangle\\ S_z|s, m \rangle = m\hbar | s, m\rangle $$ The singlet ($|s = 0, m = 0\rangle$) state is particularly relevant for quantum entanglement: measuring the spin in the z-direction of one of the particles in the singlet state, say $S_{1,z}$, provides the observer with information about the spin in the z-direction of the other particle, which is one of the eigenvalues of $S_{2, z}$.
I know that $S_{z, i}$ can be measured with a Stern-Gerlach apparatus. However, in order to reproduce entanglement in a laboratory, we have to be sure that the state is in the singlet state. This is in order to know the spin of the second particle just by measuring the spin of the first one. My question is: how can you measure the total spin, i.e. the eigenvalues of $\mathbf{S^2}$, in a lab? Or rather: how can you produce spin singlet states in a lab?
