Entanglement properties of random invariant quantum states

Entanglement properties of random multipartite quantum states which are invariant under global $\textnormal{SU}(d)$ action are investigated. The random states live in the tensor power of an irreducible representation of $\textnormal{SU}(d)$. We calculate and analyze the expectation and fluctuation of the second-order R\'enyi entanglement measure of the random invariant and near-invariant states in high dimension, and reveal the phenomenon of concentration of measure the random states exhibit. We show that with high probability a random SU($d$)-invariant state is close to being maximally entangled with respect to any bipartite cut as the dimension of individual system goes to infinity. We also show that this generic entanglement property of random SU(2)-invariant state is robust to arbitrarily finite disturbation.