  Multipartite entanglement tomography, namely the quantum Fisher information
(QFI) calculated with respect to different collective operators, allows to
fully characterize the phase diagram of the quantum Ising chain in a transverse
field with variable-range coupling. In particular, it recognizes the phase
stemming from long-range antiferromagnetic coupling, a capability also shared
by the spin squeezing. Furthermore, the QFI locates the quantum critical
points, both with vanishing and nonvanishing mass gap. In this case, we also
relate the finite-size power-law exponent of the QFI to the critical exponents
of the model, finding a signal for the breakdown of conformal invariance in the
deep long-range regime. Finally, the effect of a finite temperature on the
multipartite entanglement, and ultimately on the phase stability, is
considered. In light of the current realizations of the model with trapped ions
and of the potential measurability of the QFI, our approach yields a promising
strategy to probe long-range physics in controllable quantum systems.
