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How do I calculate the matrix representation of this part of a teleportation circuit? It must be a matrix of dimension 8.

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    $\begingroup$ Hi, Nillmer. Welcome to Quantum Computing! I recommend trying to figure it by yourself. Start here. It looks like a simple controlled-$Z$ gate to me. $\endgroup$ – Sanchayan Dutta Nov 10 '18 at 20:36
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One way to do it is to build a sort of quantum IF statement. You have in quantum computing projector operators telling you whether a qubit is 0 or 1: $$ P_0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$

$$ P_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} $$

Then we have the Z gate :

$$ Z= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$

To build the unitary for your controlled operation, we do : $$ CZ = P_0 \otimes I \otimes I + P_1 \otimes I \otimes Z $$

with $ \otimes $ the tensor product. That means, if your first qubit is in the $ |0\rangle $ state, we apply the identity operation, otherwise we apply the Z operator only on the third qubit.

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