7
$\begingroup$

Can anyone explain how the CNOT matrix below is a valid presentation for the four-qubit states that follow after?

enter image description here

|0 0> -> |0 0> 
|0 1> -> |0 1>
|1 0> -> |1 1>
|1 1> -> |1 0>

Source: Wikipedia

$\endgroup$
2
  • $\begingroup$ how what? Can you clarify what you do not find clear in the wiki page? $\endgroup$
    – glS
    Commented Oct 27, 2018 at 18:41
  • $\begingroup$ Is your confusion about the choice of ordering the basis for which of 00,01,10 and 11 go with rows/columns 1,2,3,4 of the matrix? So you know which rows/columns to put 1s vs 0s. $\endgroup$
    – AHusain
    Commented Oct 27, 2018 at 19:09

1 Answer 1

10
$\begingroup$

The one concept that I think would really help you is knowing how to turn those 4 states, $|00\rangle, |01\rangle, |10\rangle, |11\rangle$, into vectors, so that you can do the matrix multiplication.

Let me show you.

$$ \begin{align} |00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix},|01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, |10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, |11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \end{align} $$

Now if you do the matrix multplication: $\rm{CNOT} \times |00\rangle$
You will see that you will get exactly what you said, which is $|00\rangle$, and the same is true for the rest of them!

This is using the convention that $|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, and $|ab\rangle = |a\rangle \otimes |b\rangle$ where $\otimes$ is the left Kronecker product.

$\endgroup$
6
  • $\begingroup$ Thank you -- can you give me a bit more details on how you turned |11> into a vector of [0 0 0 1] (imagine is as a column vector please!) $\endgroup$
    – DrHamed
    Commented Oct 27, 2018 at 19:42
  • 2
    $\begingroup$ I gave the formula |ab> = a $\otimes$ b. So please try |11> = |1> $\otimes$ |1> ! $\endgroup$ Commented Oct 27, 2018 at 19:58
  • 2
    $\begingroup$ @DrHamed: Please look at the formula after the words "We can write out the matrix form", in this PDF: cs.cmu.edu/~odonnell/quantum15/lecture02.pdf . I believe that concludes my answer to this question. $\endgroup$ Commented Oct 27, 2018 at 20:12
  • 2
    $\begingroup$ Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer! $\endgroup$
    – DrHamed
    Commented Oct 27, 2018 at 20:13
  • 2
    $\begingroup$ Typo in 00 state, too small to make as suggested edit. $\endgroup$
    – AHusain
    Commented Oct 27, 2018 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.