  We investigate the creation of highly entangled ground states in a system of
three exchange-coupled qubits arranged in a ring geometry. Suitable magnetic
field configurations yielding approximate GHZ and exact W ground states are
identified. The entanglement in the system is studied at finite temperature in
terms of the mixed-state tangle tau. By adapting a steepest-descent
optimization algorithm we demonstrate that tau can be evaluated efficiently and
with high precision. We identify the parameter regime for which the equilibrium
entanglement of the tripartite system reaches its maximum.
