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 [Carlen, Loss and Lieb, Meth. and Appl. of Analysis, 13 (2006), no. 1, 1–17], we can write combinatorial formulas for quantifying the entanglement of permutation invariant basis states. When applying the geometric measure to permutation invariant states with nonnegative coefficients, we show that the overlap with a product state is maximized by a tensor product of the same single-party state. This extends some observations in [Hayashi et al., Phys. Rev. A 77, 012104 (2008)].