Find the $|$

$$

$$ output state. Here U $=|00|+$ e

i$|11|$ is a relative phase gate.

$($b$)$ Show that the relative phase applied by U can be calculated as $=$ cos$1$

$[$p$0$ p$1$

$]$ where

p$0$ and p$1$ are the probabilities finding the $|$

$$

$$ state in $|0$ and $|1$ respectively.