Stochastic behavior of outcome of Schur-Weyl duality measurement
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 $ρ_{mix,n,l}$ 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 $|Ξ_{n,l}\rangle$, 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 $ρ_{mix,k,l}$ and the Dicke state $| Ξ_{n-k,m-l} \rangle$. We derive various types of asymptotic distribution including a kind of central limit theorem when $n$ goes to infinity.