Problem:
Use the matrix form of \(\otr{-}{-} \otimes I\)
and apply that to \(\ksi = \isqt{1} \parenl{ \kt{00} + \kt{11} } \).
Solution:
$$
\otr{-}{-} \otimes I
\eql
\left( \mat{\isqt{1} \\ -\isqt{1}} \mat{\isqt{1} & -\isqt{1}} \right)
\otimes I
\eql
\mat{
\frac{1}{2} I & -\frac{1}{2} I\\
-\frac{1}{2} I & \frac{1}{2} I\\
}
\eql
\mat{
\frac{1}{2} & 0 & -\frac{1}{2} & 0 \\
0 & \frac{1}{2} & 0 & -\frac{1}{2} \\
-\frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} \\
}
$$
Then,
$$
(\otr{-}{-} \otimes I) \ksi
\eql
\mat{
\frac{1}{2} & 0 & -\frac{1}{2} & 0 \\
0 & \frac{1}{2} & 0 & -\frac{1}{2} \\
-\frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} \\
}
\mat{\isqt{1} \\ 0 \\ 0 \\ \isqt{1}}
\eql
\isqt{1} \mat{ \frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ \frac{1}{2} }
\eql
\isqt{1} \parenl{ \kt{-} \otimes \kt{-} }
$$
The last step comes from
$$
\kt{-} \otimes \kt{-}
\eql
\left( \mat{ \isqt{1} \\ -\isqt{1} } \right)
\otimes
\left( \mat{ \isqt{1} \\ -\isqt{1} } \right)
\eql
\mat{ \frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ \frac{1}{2} }
$$
Problem:
What is the result of \((I\otimes I \otimes X) (\kt{01}(\beta\kt{0} + \alpha\kt{1}))\)?
Solution:
$$\eqb{
(I\otimes I \otimes X)
(\kt{01}(\beta\kt{0} + \alpha\kt{1}))
& \eql &
(I\otimes I \otimes X)
\parenl{ \kt{0} \otimes \kt{1} \otimes (\beta\kt{0} +
\alpha\kt{1}) } \\
& \eql &
I\kt{0} \otimes I\kt{1} \otimes X(\beta\kt{0} +
\alpha\kt{1}) \\
& \eql &
\kt{0} \otimes \kt{1} \otimes \alpha\kt{0} +
\beta\kt{1}) \\
& \eql &
\alpha \kt{010} + \beta\kt{011}
}$$
Recall: \(X\) is one of the four Pauli operators:
$$
X (\alpha\kt{0} + \beta\kt{1})
\eql
\mat{0 & 1\\ 1 & 0} \mat{\alpha \\ \beta}
\eql
\mat{\beta \\ \alpha}
\eql
\beta\kt{0} + \alpha\kt{1}
$$
Problem:
Show that
$$
(H \otimes I \otimes I)
\parenl{ \alpha\kt{000} + \beta\kt{101} }
\eql
\isqt{1} \parenl{ \kt{00} \otimes (\alpha\kt{0} + \beta\kt{1})
+
\kt{10} \otimes (\alpha\kt{0} - \beta\kt{1}) }
$$
Solution:
$$\eqb{
(H \otimes I \otimes I)
\parenl{ \alpha\kt{000} + \beta\kt{101} }
& \eql &
\alpha H\kt{0}\kt{0}\kt{0} + \beta H\kt{1}\kt{0}\kt{1} \\
& \eql &
\alpha \isqt{1} (\kt{0} + \kt{1}) \kt{0}
+ \beta \isqt{1} (\kt{0} + \kt{1}) \kt{0}\kt{1} \\
& \eql &
\isqt{1} \parenl{ \alpha\kt{000} + \alpha\kt{100}
+ \beta\kt{001} - \beta\kt{101} } \\
& \eql &
\isqt{1} \parenl{ \alpha\kt{000} + \beta\kt{001}
+ \alpha\kt{100} - \beta\kt{101} } \\
& \eql &
\isqt{1} \parenl{
\kt{00} \otimes \alpha\kt{0} + \kt{00} \otimes \beta\kt{1}
+ \kt{10} \otimes \alpha\kt{0} - \kt{10} \otimes \beta\kt{1} } \\
& \eql &
\isqt{1} \parenl{
\kt{00} \otimes (\alpha\kt{0} + \beta\kt{1})
+ \kt{10} \otimes (\alpha\kt{0} - \beta\kt{1}) } \\
}$$