General method for obtaining the energy minimum of spin Hamiltonians for separable states

We present a general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. For ferromagnetic Ising and Ising-like models with nearest-neighbor interactions on lattices of any dimension and on a fully connected graph in an external field, this minimum is given by a compact analytic formula involving the quantum Fisher information. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the direct extraction of both the quantum Fisher information and the fidelity from correlation measurements on the ground states of suitably engineered spin models.