- #1
James Jackson
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I'm just looking at another quantum computation question. It is stated like so:
The operators Y and Z on [tex]C^2[/tex] are defined by:
[tex]Y|0\rangle =i|1\rangle ; Y|1\rangle = -i|0\rangle[/tex]
[tex]Z|0\rangle = |0\rangle ; Z|1\rangle = -|1\rangle[/tex]
Write Z in diagonal form
Write Y in Dirac form with respect to the basis [tex]\{ 0\rangle , |1\rangle\}[/tex]
Now, I'm confusing myself something silly. I'm under the impression that the diagonal form of an operator is given by:
[tex]A=\sum \lambda_{n}|n\rangle\langle n|[/tex]
where [tex]|n\rangle[/tex] are the eigenvectors and [tex]\lambda_n[/tex] are the eigenvalues of A.
But I would also take this to be the Dirac form, so I'm clearly missing something.
The eigenvalues of Z are clearly [tex]\{1,-1\}[/tex] with eigenvectors [tex]\{ |0\rangle ,|1\rangle\}[/tex], so the diagonal form is [tex]Z=|0\rangle\langle 0|-|1\rangle\langle 1|[/tex].
I suppose my question breaks down to 'What is meant by the Dirac form of an operator?'
Any hints?
Edited to remove me being stupid and working out eigenvectors incorrectly.
Edit: Or, by Dirac form of an operator, do they mean the matrix representation which, for Y, is given by:
[tex]Y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right)[/tex]
The operators Y and Z on [tex]C^2[/tex] are defined by:
[tex]Y|0\rangle =i|1\rangle ; Y|1\rangle = -i|0\rangle[/tex]
[tex]Z|0\rangle = |0\rangle ; Z|1\rangle = -|1\rangle[/tex]
Write Z in diagonal form
Write Y in Dirac form with respect to the basis [tex]\{ 0\rangle , |1\rangle\}[/tex]
Now, I'm confusing myself something silly. I'm under the impression that the diagonal form of an operator is given by:
[tex]A=\sum \lambda_{n}|n\rangle\langle n|[/tex]
where [tex]|n\rangle[/tex] are the eigenvectors and [tex]\lambda_n[/tex] are the eigenvalues of A.
But I would also take this to be the Dirac form, so I'm clearly missing something.
The eigenvalues of Z are clearly [tex]\{1,-1\}[/tex] with eigenvectors [tex]\{ |0\rangle ,|1\rangle\}[/tex], so the diagonal form is [tex]Z=|0\rangle\langle 0|-|1\rangle\langle 1|[/tex].
I suppose my question breaks down to 'What is meant by the Dirac form of an operator?'
Any hints?
Edited to remove me being stupid and working out eigenvectors incorrectly.
Edit: Or, by Dirac form of an operator, do they mean the matrix representation which, for Y, is given by:
[tex]Y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right)[/tex]
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