Why the CNOT gate matrix is a valid representation for four-qubit states?

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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






quantum-gate controlled-gates







share|improve this question























  • how what? Can you clarify what you do not find clear in the wiki page?
    – glS
    3 hours ago










  • Thanks, just fixed the post!
    – DrHamed
    3 hours ago










  • 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.
    – AHusain
    3 hours ago














up vote
1
down vote

favorite












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






quantum-gate controlled-gates







share|improve this question























  • how what? Can you clarify what you do not find clear in the wiki page?
    – glS
    3 hours ago










  • Thanks, just fixed the post!
    – DrHamed
    3 hours ago










  • 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.
    – AHusain
    3 hours ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











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






quantum-gate controlled-gates







share|improve this question















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






quantum-gate controlled-gates




quantum-gate controlled-gates






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 2 hours ago









bytebuster

2521115




2521115










asked 4 hours ago









DrHamed

516




516











  • how what? Can you clarify what you do not find clear in the wiki page?
    – glS
    3 hours ago










  • Thanks, just fixed the post!
    – DrHamed
    3 hours ago










  • 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.
    – AHusain
    3 hours ago
















  • how what? Can you clarify what you do not find clear in the wiki page?
    – glS
    3 hours ago










  • Thanks, just fixed the post!
    – DrHamed
    3 hours ago










  • 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.
    – AHusain
    3 hours ago















how what? Can you clarify what you do not find clear in the wiki page?
– glS
3 hours ago




how what? Can you clarify what you do not find clear in the wiki page?
– glS
3 hours ago












Thanks, just fixed the post!
– DrHamed
3 hours ago




Thanks, just fixed the post!
– DrHamed
3 hours ago












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.
– AHusain
3 hours ago




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.
– AHusain
3 hours ago










1 Answer
1






active

oldest

votes

















up vote
3
down vote



accepted










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



Let me show you.



$$
beginalign
|00rangle = beginbmatrix 1 \ 0 \ 0 \ 0 endbmatrix,|01rangle = beginbmatrix 0 \ 1 \ 0 \ 0 endbmatrix, |10rangle = beginbmatrix 0 \ 0 \ 1 \ 0 endbmatrix, |11rangle = beginbmatrix 0 \ 0 \ 0 \ 1 endbmatrix
endalign
$$



Now if you do the matrix multplication: $rmCNOT times |00rangle$

You will see that you will get exactly what you said, which is $|00rangle$, and the same is true for the rest of them!



This is using the convention that $|0rangle = beginbmatrix 1 \ 0 endbmatrix$ and $|1rangle = beginbmatrix 0 \ 1 endbmatrix$, and $|abrangle = |arangle otimes |brangle$ where $otimes$ is the left Kronecker product.






share|improve this answer






















  • 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!)
    – DrHamed
    2 hours ago






  • 2




    I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
    – user1271772
    2 hours ago






  • 2




    @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.
    – user1271772
    2 hours ago






  • 1




    Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
    – DrHamed
    2 hours ago






  • 2




    Typo in 00 state, too small to make as suggested edit.
    – AHusain
    1 hour ago










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
3
down vote



accepted










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



Let me show you.



$$
beginalign
|00rangle = beginbmatrix 1 \ 0 \ 0 \ 0 endbmatrix,|01rangle = beginbmatrix 0 \ 1 \ 0 \ 0 endbmatrix, |10rangle = beginbmatrix 0 \ 0 \ 1 \ 0 endbmatrix, |11rangle = beginbmatrix 0 \ 0 \ 0 \ 1 endbmatrix
endalign
$$



Now if you do the matrix multplication: $rmCNOT times |00rangle$

You will see that you will get exactly what you said, which is $|00rangle$, and the same is true for the rest of them!



This is using the convention that $|0rangle = beginbmatrix 1 \ 0 endbmatrix$ and $|1rangle = beginbmatrix 0 \ 1 endbmatrix$, and $|abrangle = |arangle otimes |brangle$ where $otimes$ is the left Kronecker product.






share|improve this answer






















  • 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!)
    – DrHamed
    2 hours ago






  • 2




    I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
    – user1271772
    2 hours ago






  • 2




    @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.
    – user1271772
    2 hours ago






  • 1




    Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
    – DrHamed
    2 hours ago






  • 2




    Typo in 00 state, too small to make as suggested edit.
    – AHusain
    1 hour ago














up vote
3
down vote



accepted










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



Let me show you.



$$
beginalign
|00rangle = beginbmatrix 1 \ 0 \ 0 \ 0 endbmatrix,|01rangle = beginbmatrix 0 \ 1 \ 0 \ 0 endbmatrix, |10rangle = beginbmatrix 0 \ 0 \ 1 \ 0 endbmatrix, |11rangle = beginbmatrix 0 \ 0 \ 0 \ 1 endbmatrix
endalign
$$



Now if you do the matrix multplication: $rmCNOT times |00rangle$

You will see that you will get exactly what you said, which is $|00rangle$, and the same is true for the rest of them!



This is using the convention that $|0rangle = beginbmatrix 1 \ 0 endbmatrix$ and $|1rangle = beginbmatrix 0 \ 1 endbmatrix$, and $|abrangle = |arangle otimes |brangle$ where $otimes$ is the left Kronecker product.






share|improve this answer






















  • 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!)
    – DrHamed
    2 hours ago






  • 2




    I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
    – user1271772
    2 hours ago






  • 2




    @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.
    – user1271772
    2 hours ago






  • 1




    Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
    – DrHamed
    2 hours ago






  • 2




    Typo in 00 state, too small to make as suggested edit.
    – AHusain
    1 hour ago












up vote
3
down vote



accepted







up vote
3
down vote



accepted






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



Let me show you.



$$
beginalign
|00rangle = beginbmatrix 1 \ 0 \ 0 \ 0 endbmatrix,|01rangle = beginbmatrix 0 \ 1 \ 0 \ 0 endbmatrix, |10rangle = beginbmatrix 0 \ 0 \ 1 \ 0 endbmatrix, |11rangle = beginbmatrix 0 \ 0 \ 0 \ 1 endbmatrix
endalign
$$



Now if you do the matrix multplication: $rmCNOT times |00rangle$

You will see that you will get exactly what you said, which is $|00rangle$, and the same is true for the rest of them!



This is using the convention that $|0rangle = beginbmatrix 1 \ 0 endbmatrix$ and $|1rangle = beginbmatrix 0 \ 1 endbmatrix$, and $|abrangle = |arangle otimes |brangle$ where $otimes$ is the left Kronecker product.






share|improve this answer














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



Let me show you.



$$
beginalign
|00rangle = beginbmatrix 1 \ 0 \ 0 \ 0 endbmatrix,|01rangle = beginbmatrix 0 \ 1 \ 0 \ 0 endbmatrix, |10rangle = beginbmatrix 0 \ 0 \ 1 \ 0 endbmatrix, |11rangle = beginbmatrix 0 \ 0 \ 0 \ 1 endbmatrix
endalign
$$



Now if you do the matrix multplication: $rmCNOT times |00rangle$

You will see that you will get exactly what you said, which is $|00rangle$, and the same is true for the rest of them!



This is using the convention that $|0rangle = beginbmatrix 1 \ 0 endbmatrix$ and $|1rangle = beginbmatrix 0 \ 1 endbmatrix$, and $|abrangle = |arangle otimes |brangle$ where $otimes$ is the left Kronecker product.







share|improve this answer














share|improve this answer



share|improve this answer








edited 1 hour ago

























answered 3 hours ago









user1271772

4,984233




4,984233











  • 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!)
    – DrHamed
    2 hours ago






  • 2




    I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
    – user1271772
    2 hours ago






  • 2




    @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.
    – user1271772
    2 hours ago






  • 1




    Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
    – DrHamed
    2 hours ago






  • 2




    Typo in 00 state, too small to make as suggested edit.
    – AHusain
    1 hour ago
















  • 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!)
    – DrHamed
    2 hours ago






  • 2




    I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
    – user1271772
    2 hours ago






  • 2




    @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.
    – user1271772
    2 hours ago






  • 1




    Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
    – DrHamed
    2 hours ago






  • 2




    Typo in 00 state, too small to make as suggested edit.
    – AHusain
    1 hour ago















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!)
– DrHamed
2 hours ago




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!)
– DrHamed
2 hours ago




2




2




I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
– user1271772
2 hours ago




I gave the formula |ab> = a $otimes$ b. So please try |11> = |1> $otimes$ |1> !
– user1271772
2 hours ago




2




2




@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.
– user1271772
2 hours ago




@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.
– user1271772
2 hours ago




1




1




Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
– DrHamed
2 hours ago




Great, I have the Michael Nielsen book, it completely skipped this step, which I found it frustrating. Thanks again for this detailed answer!
– DrHamed
2 hours ago




2




2




Typo in 00 state, too small to make as suggested edit.
– AHusain
1 hour ago




Typo in 00 state, too small to make as suggested edit.
– AHusain
1 hour ago

















 

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