Stochastic behavior of outcome of Schur-Weyl duality measurement

Understand

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.