Spin squeezing from bilinear spin-spin interactions: Two simple theorems

We prove two simple theorems about squeezing induced by bilinear spin-spin interactions that conserve spin parity—including a vast majority of quantum spin models implemented by state-of-the-art quantum simulators. In particular we show that squeezing captures the first form of quantum correlations which are produced (1) at equilibrium, by adiabatically turning on the spin-spin interactions starting from a factorized state aligned with an external, arbitrary field; and (2) away from equilibrium, by evolving unitarily the same state with the interacting Hamiltonian.