Ex5 5

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 5.5}$

A vector $\ket\psi$ is stabilized by an operator $U$ if $U\ket\psi = \ket\psi$. Find the set of vectors stabilized by

a) the Pauli operator $X$,

b) the Pauli operator $Y$,

c) the Pauli operator $Z$,

d) $X\otimes X$,

e) $Z\otimes X$,

f) $C_{not}$.