Ex4 13

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.13}$

a) Let $O$ be any observable specifying a measurement of an $n$-qubit system. Suppose that after measuring $\ket\psi$ according to $O$ we obtain $\ket\phi$. Show that if we now measure $\ket\phi$ according to $O$ we simply obtain $\ket\phi$ again, with certainty.

b) Reconcile the result of part a) with the fact that for most observables $O$ it is not true that $O^2 = O$.