Ex4 2

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.2}$

Write the following operators in bra/ket notation

a) The Hadamard operator $H = \left(\begin{array}{cc}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\\frac{1}{\sqrt 2} & -\frac{1}{\sqrt 2}\end{array}\right)$.

b) $X = \left(\begin{array}{cc}0 & 1 \\1 & 0 \end{array}\right)$.

c) $Y = \left(\begin{array}{cc}0 & 1 \\-1 & 0 \end{array}\right)$.

d) $Z = \left(\begin{array}{cc}1 & 0 \\0 & -1 \end{array}\right)$.

e) $\left(\begin{array}{cccc}23 & 0 & 0 & 0 \\0 & -5 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 9 \end{array}\right)$.

f) $X\otimes X$.

g) $X\otimes Z$.

h) $H\otimes H$.

i) The projection operators $P_1: V \to S_1$ and $P_2: V \to S_2$, where $S_1$ is spanned by $\{ \ket +\ket +, \ket -\ket - \}$ and $S_2$ is spanned by $\{ \ket +\ket -, \ket -\ket + \}$.