Higher-order Spatially Separated Correlations and Quantum Magic in the Long-range Transverse Field Ising Model

Long-range interacting quantum systems exhibit interesting phenomena beyond the nearest-neighbours regime. Such interactions can reach further separated particles and change the behaviour of many-body correlations. Here, we analyse their structure in the antiferromagnetic long-range interaction transverse field Ising model with power-law interactions decaying as 1/r^α. We examine the spatial behaviour of basis-independent total multipartite correlations, called sector lengths S_k, of its ground state in the vicinity of critical points for different interaction exponents α and fixed system size. These correlation functions were shown to indicate entanglement and the existence of local realistic models, allowing us to make claims about the Bell nonlocality and entanglement in the system. We further investigate them in the context of non-stabiliserness, often referred to as quantum magic, which holds information on their classical simulability and usefulness for quantum computation. We evaluate R{é}nyi stabiliser entropy near and outside the criticality and show that cumulative magic, up to the critical point, increases with α for all of the examined system sizes. Our results provide new insights into the interplay between the long-range interactions, observable non-classical phenomena and their classical simulability.