Entanglement and group symmetries: stabilizer, symmetric and antisymmetric states

We study distance-like entanglement measures of multipartite states with certain symmetries. Using group averaging we show general conditions for which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equal. We then consider specific sets of states important in quantum information and many-body physics. We show equivalence of these measures for all stabilizer states, symmetric basis and antisymmetric basis states. We also calculate the explicit value of these measures for symmetric and antisymmetric basis states, indicating that antisymmetric states are generally more entangled.