What is this state as a matrix? In QM I have the state $\lvert 00 \rangle \langle 00 \rvert$. Can anyone tell me what this would look like as a matrix? I know that
$$ \lvert 00 \rangle = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}. $$
 A: $|00\rangle\langle00|$ is a unit matrix element on $\mathbb{C}^2\otimes \mathbb{C}^2$ space. In the same way that $\hat{\boldsymbol{i}}$ or $\hat{\boldsymbol{x}}$ is a unit vector. For example if:
$$M=|00\rangle\langle00|$$
Then this could be represented in matrix form as:
$$M=\begin{pmatrix}
1 & 0 & 0 & 0\\ 
0 & 0 & 0 & 0\\ 
0 & 0 &  0& 0\\ 
0 & 0 & 0 & 0
\end{pmatrix}$$
In the same way as, if:
$$v = \hat{\boldsymbol{i}}$$
Then,
$$v =  \begin{pmatrix}
1 \\ 
0 \\ 
\end{pmatrix}$$
So the last comment is not true. A ket element represents a vector (not a matrix):
$$|00\rangle\ = \begin{pmatrix}
1 \\ 
0 \\
0 \\
0 \\ 
\end{pmatrix} \neq \begin{pmatrix} 1&1 \\0&0\end{pmatrix}$$
In general:


*

*a ket (e.g. $|00\rangle$) represents a vector

*a bra (e.g. $\langle00|$) represents a co-vetor, dual-vector or one-form (which all mean the same thing in QM)

*a ket-bra (e.g. $|00\rangle\langle00|$) a matrix element

*a bra-ket (e.g. $\langle00||00\rangle = \langle00|00\rangle$) a scalar quantity

A: Keep in mind that $|a\rangle=\langle a|^*$. This means that $\langle 00|$ will be the reverse and adjoint of $|00\rangle$. So you'll get $\begin{pmatrix} 1&0 \\1&0\end{pmatrix}$. The product will give you a new $4\times4$ matrix.
A: I've never seen this being claimed as true...
$$ \lvert 00 \rangle = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} $$
$\lvert 00 \rangle$ is a shorthand for a kronecker product, $\lvert 0 \rangle \otimes \lvert 0 \rangle$, which should be represented by a vector, not a matrix.
