Nielsen and Chuang -- Chapter 6

Updated on January 11, 2019

(6.2) Show that the operation $(2 \ket{\psi}\bra{\psi} - I)$ (where $\ket{\psi}$ is the equally weighted superposition of states) applied to general state $\sum_k \alpha_k \ket{k}$ produces

where $\langle \alpha \rangle \equiv \sum_k \alpha_k / N$ is the mean value of $\alpha_k$.