  We study various distance-like entanglement measures of multipartite states
under certain symmetries. Using group averaging techniques we provide
conditions under which the relative entropy of entanglement, the geometric
measure of entanglement and the logarithmic robustness are equivalent. We
consider important classes of multiparty states, and in particular show that
these measures are equivalent for all stabilizer states, symmetric basis and
antisymmetric basis states. We rigorously prove a conjecture that the closest
product state of permutation symmetric states can always be chosen to be
permutation symmetric. This allows us to calculate the explicit values of
various entanglement measures for symmetric and antisymmetric basis states,
observing that antisymmetric states are generally more entangled. We use these
results to obtain a variety of interesting ensembles of quantum states for
which the optimal LOCC discrimination probability may be explicitly determined
and achieved. We also discuss applications to the construction of optimal
entanglement witnesses.
