I'm having an issue with some outer & Kronecker products where I am doing two different processes which should result in the same answer, but I'm getting a different answer for each. Can anyone tell me where I'm going wrong?
I have the following:
$A = \begin{bmatrix}1\\0\end{bmatrix}$
$B = \begin{bmatrix}1\\0\end{bmatrix}$
$H = \begin{bmatrix}1&1\\1&-1\end{bmatrix}*\frac{1}{\sqrt{2}}$
The first process is: $Vec((H*A)\otimes B)$
$H*A = \begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$
Taking the outer product with the $B$ vector:
$\begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix} * \begin{bmatrix}1&0\end{bmatrix} = \begin{bmatrix}\frac{1}{\sqrt{2}}&0\\\frac{1}{\sqrt{2}}&0\end{bmatrix}$
Vectorizing that, we get:
$\begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\\0\end{bmatrix}$
The second process is: $(H\otimes I)*Vec(A\otimes B)$
$H\otimes I = \begin{bmatrix}1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\end{bmatrix}*\frac{1}{\sqrt{2}}$
$Vec(A\otimes B) = \begin{bmatrix}1\\0\\0\\0\end{bmatrix}$
Multiplying them together we get:
$\begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\\frac{1}{\sqrt{2}}\\0\end{bmatrix}$
Can anyone see where I've mis-stepped? I'm pretty sure the second process comes up with the correct answer and that the first process is somehow incorrect.
I'm trying to do the first step to making a Bell state to give some context. The next step would be the CNOT gate.
Thanks!!
EDIT: It turns out that I'm having the same problem as this person was: Kronecker product and outer product confusion
You said:
But in your case $B=\begin{bmatrix}1\\ 0\end{bmatrix},$ so it should have been
$$\begin{bmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix} \otimes \begin{bmatrix}1\\ 0\end{bmatrix} = \begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\\frac{1}{\sqrt{2}}\\0\end{bmatrix}.$$
Vectorization converts matrix to colum vector, but in your case $(H*A)\otimes B$ is already a vector, hence $Vec((H*A)\otimes B)=(H*A)\otimes B.$