$(20$ points$)$ Consider the Hadamard matrix

H $=$

$1$

$$

$2$

$11$

$11$

$(6)$

and the normalized state

$|\backslash$psi $=$

cos$(\backslash$theta $)$

sin$(\backslash$theta $)$

$,(7)$

calculate the variance

VH$(|\backslash$psi $)=\backslash$psi $|$ H

$2$

$|\backslash$psi $\backslash$psi $|$ H $|\backslash$psi $$

$2$

$,$