  Interconversions of $W$ and Greenberger-Horne-Zeilinger states in various
physical systems are lately attracting considerable attention. We address this
problem in the fairly general physical setting of qubit arrays with long-ranged
(all-to-all) Ising-type qubit-qubit interaction, which are simultaneously acted
upon by transverse Zeeman-type global control fields. Motivated in part by a
recent Lie-algebraic result that implies state-to-state controllability of such
a system for an arbitrary pair of states that are invariant with respect to
qubit permutations, we present a detailed investigation of the
state-interconversion problem in the three-qubit case. The envisioned
interconversion protocol has the form of a pulse sequence that consists of two
instantaneous (delta-shaped) control pulses, each of them corresponding to a
global qubit rotation, and an Ising-interaction pulse of finite duration
between them. Its construction relies heavily on the use of the
(four-dimensional) permutation-invariant subspace (symmetric sector) of the
three-qubit Hilbert space. In order to demonstrate the viability of the
proposed state-interconversion scheme, we provide a detailed analysis of the
robustness of the underlying pulse sequence to systematic errors, i.e.
deviations from the optimal values of its five characteristic parameters.
