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We point out that a geometric measure of quantum entanglement is related tothe matrix permanent when restricted to permutation invariant states. Thisconnection allows us to interpret the permanent as an angle between vectors. Byemploying a recently introduced permanent inequality by Carlen, Loss and Lieb,we can prove explicit formulas of the geometric measure for permutationinvariant basis states in a simple way.

Matrix permanent and quantum entanglement of permutation invariant states