Can anyone explain how the CNOT matrix below is a valid presentation for the four-qubit states that follow after?
|0 0> -> |0 0>
|0 1> -> |0 1>
|1 0> -> |1 1>
|1 1> -> |1 0>
Source: Wikipedia
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$\begingroup$ how what? Can you clarify what you do not find clear in the wiki page? $\endgroup$– glS ♦Oct 27, 2018 at 18:41
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$\begingroup$ 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. $\endgroup$– AHusainOct 27, 2018 at 19:09
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1 Answer
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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.
answered Oct 27, 2018 at 19:10
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$\begingroup$ 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!) $\endgroup$– DrHamedOct 27, 2018 at 19:42
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2$\begingroup$ I gave the formula |ab> = a $\otimes$ b. So please try |11> = |1> $\otimes$ |1> ! $\endgroup$ Oct 27, 2018 at 19:58
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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. $\endgroup$ Oct 27, 2018 at 20:12
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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! $\endgroup$– DrHamedOct 27, 2018 at 20:13
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2$\begingroup$ Typo in 00 state, too small to make as suggested edit. $\endgroup$– AHusainOct 27, 2018 at 20:53