Outer Product Intution

-1

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".

Binshumesh sachan

Posted 2020-10-15T17:09:40.487

Reputation: 95

Question was closed 2020-10-19T09:44:42.550

A related answer.

– Davit Khachatryan – 2020-10-15T18:12:35.190

Answers

3

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.

KAJ226

Posted 2020-10-15T17:09:40.487

Reputation: 6 322

Hi Binshumesh, adding to the well-written answer above, you can intuitively think about the 'bra' in the outer product (operator) as the part combining (via the inner product) with the vector input to the operator to give the numeric coefficient of the output vector, and the 'ket' in the operator being the label of the output vector itself. So (|0><1|)|1> =( |0>)(<1|1>) = (<1|1>)(|0>) = (1)(|0>) = |0>. – Dhruv B – 2020-10-15T18:21:37.580

Thanks for your explanation. – Binshumesh sachan – 2020-10-19T08:23:07.560