  We focus on the measurement defined by the decomposition based on Schur-Weyl
duality on $n$ qubits. As the first setting, we discuss the asymptotic behavior
of the measurement outcome when the state is given as the permutation mixture
$\rho_{mix,n,l}$ of the state $| 1^{l} \, 0^{n-l} \rangle := | 1
\rangle^{\otimes l} \otimes |0\rangle^{\otimes (n-l)}$. In contrast, when the
state is given as the Dicke state $|\Xi_{n,l}\rangle$, the measurement outcome
takes one deterministic value. These two cases have completely different
behaviors. As the second setting, we study the case when the state is given as
the tensor product of the permutation mixture $\rho_{mix,k,l}$ and the Dicke
state $| \Xi_{n-k,m-l} \rangle$. We derive various types of asymptotic
distribution including a kind of central limit theorem when $n$ goes to
infinity.
