Verification of phased Dicke states

Dicke states are typical examples of quantum with genuine multipartite entanglement. They valuable resources in many information processing tasks, including multiparty communication and metrology. Phased a generalization include antisymmetric basis as special example. These useful atomic molecular physics besides processing. Here we propose practical efficient protocols based on adaptive local projective measurements for verifying all phased states, $W$ qudit states. To verify any $n$-partite state within infidelity $\ensuremath{\epsilon}$ significance level $\ensuremath{\delta}$, the number tests required is only $O(n{\ensuremath{\epsilon}}^{\ensuremath{-}1}ln{\ensuremath{\delta}}^{\ensuremath{-}1})$, which linear $n$ exponentially more than traditional tomographic approaches. In case can be further reduced to $O(\sqrt{n}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\epsilon}}^{\ensuremath{-}1}ln{\ensuremath{\delta}}^{\ensuremath{-}1})$. Moreover, construct an optimal protocol state; decreases (rather increases) monotonically $n$. This known nonstabilizer