Matrix permanent and quantum entanglement of permutation invariant states

We point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality by Carlen et al. [Methods Appl. Anal.13, 1 (2006)], we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way.