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New to quantum and ran into the block-encoding. Having a bit of trouble understanding $|0\rangle \otimes I$.
$|0\rangle$ is just a vector but $I$ is an $n$ by $n$ matrix? Not clear how vector can be tensorproducted with a matrix? I know I am missing something here.
Any help could be appreciated.
This is probably best done with an example. Let's consider a $4\times 4$ matrix $U$ which acts on two qubits. The $|0\rangle\otimes I$ is equivalent to $$ \left(\begin{array}{c} 1 \\ 0 \end{array}\right)\otimes\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}\right) $$ (if you don't know where this comes from, go back to the definition of the tensor product). This is a $4\times 2$ matrix, meaning it's the right size for you to pre-multiply by $U$. Similarly, $$ \langle 0|\otimes I\equiv \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right) $$ so that, overall your matrix $A$ comes out as $2\times 2$ (it's the action of what happens to the second qubit if the first qubit starts and ends in state $|0\rangle$).
