Generation of many-body entanglement in long-range interacting systems

The existence of long-ranged interactions generally complicates the description of many-body systems. However, in the limit where the interactions become infinitely long-ranged---i.e. independent of distance---the emergence of extra conserved quantities typically makes the behavior quite simple. Such infinite-ranged interactions are often assumed in the description of experiments aiming to produce large scale entangled states, for instance via spin-squeezing, but of course ``infinite'' in this context is an idealization. We consider the generation of entanglement in Ising models with long (but not infinite) ranged interactions, which are relevant to the description of a variety of quantum information/simulation platforms including trapped ions, polar molecules, Rydberg atoms, and nitrogen vacancy centers in diamond. We demonstrate that there exists a notion of sufficiently long-ranged interactions, for which the scaling of entanglement with system size expected from the infinite-range idealization is completely unmodified. Our results have direct applications to experimental protocols aiming to achieve quantum-enhanced metrology.