$5\mathrm{.}1.$ Consider the following state vector:

$|$:

$($A$)$ Is the state normalized?

$($B$)$ What is the probability that the system is found to be in state $|0|$ if $Z$ is measured?

$($C$)$ Wite down the density operator.

$($D$)$ Find the density matrix in the $\{$:$\{|O_{?}),|I$:$)$ basis, and show that $Tr()=1.$

$5\mathrm{.}2.$ Consider the state

$|$:

Is this state normalized? Is $=|$: a density operator?

$5\mathrm{.}3.$ Let

$|$:

$($A$)$ Wite down the density matrix in the $\{|0|I\},$

Solve these $4$ questions in handwritten