  We state and prove two theorems about the ground state of magnetic systems
described by very general Heisenberg-type models and discuss their implications
for magnetic neutron scattering. The first theorem states that two models
cannot have the same correlator without sharing the corresponding ground
states. According to the second theorem, an $N$-qubit wave function cannot
reproduce the correlators of a given system unless it represents a true ground
state of that system. We discuss the implications for neutron scattering
inverse problems. We argue that the first theorem provides a framework to
understand neutron-based Hamiltonian learning. Furthermore, we propose a
variational approach to quantum magnets based on the second theorem where a
representation of the wave function (held, for instance, in a neural network or
in the qubit register of a quantum processor) is optimised to fit experimental
neutron-scattering data directly, without the involvement of a model
Hamiltonian.
