Outer Product Intution

Question

Please, help me understand this statement. The outer product notation for matrices also gives an intuitive input-output relation for them. For instance, the matrix |0⟩ ⟨1| + |1⟩ ⟨0| can be read as "output 0 when given a 1 and output 1 when given a 0".

Answer 1

Note that

$$ |0\rangle\langle 1| = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $$ and similarly

$$ |1\rangle\langle 0| = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix}= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} $$ and therefore $$ X = |0\rangle\langle 1| + |1\rangle\langle 0| = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$

Thus you can see that $X|0\rangle = |1\rangle$ and $X|1\rangle = |0\rangle$. So giving the input is the state $|0\rangle$ then the output is the state $|1\rangle$ and vice versa.