Ex11 10

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 11.10}$

a) Show that for all $i$ and $j$ and for all orthonormal $\ket{c_1}\ne\ket{c_2}\in C$,

(1)
\begin{align} \bra {c_1} Z_j^\dagger Y_i \ket {c_2} &=& 0 \\ \bra {c_1} I Y_i \ket {c_2} &=& 0, \end{align}

b) Show that for all $j \ne i$ and for all orthonormal $\ket{c_1}\ne\ket{c_2}\in C$,

(2)
\begin{align} \bra {c_1} Y_j^\dagger Y_i \ket {c_2} = 0. \end{align}