- #1

James Jackson

- 162

- 0

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]

Last edited: