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## Main Question or Discussion Point

in particular, i wonder if the trasposition super operator is basis independent or not.

We can always write an operator W as

[itex]\hat{W}=\sum_{i,j} c_{i,j} |i\rangle\langle j| [/itex]

and for the transposed we obtain

[itex]\hat{W}^T=\sum_{i,j} c_{j,i} |i\rangle\langle j| [/itex]

we obtain a relation true for each \psi,\phi

[itex]\langle\psi|\hat{W}^T|\phi\rangle=\langle\phi|\hat{W}|\psi\rangle [/itex] (1)

Now let's write the transposition super-operator as

[itex]\Lambda[\hat{W}]=\sum_{i,j} |i\rangle\langle j| \hat{W} |i\rangle\langle j| = \hat{W}^T[/itex]

Now seems that the

I have to admit it because the formula (1) is true for each vector and don't depend from the basis.

But that's sound strange to me, because if I have an operator, it can always be diagonal in a certain basis, and so it is equal to its transposition.

We can always write W diagonal in a certain basis [itex]\{|\tilde{k}>\}[/itex] not neccesary equal to [itex]\{|i>\}[/itex]:

[itex]

\hat{W}=\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|[/itex]

and from here do i get

[itex]

\hat{W}^T=(\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|)^T=(\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|=\hat{W}[/itex]

Where i'm wrong?

Can I write every super operator in a representation that is independent from the basis?

We can always write an operator W as

[itex]\hat{W}=\sum_{i,j} c_{i,j} |i\rangle\langle j| [/itex]

and for the transposed we obtain

[itex]\hat{W}^T=\sum_{i,j} c_{j,i} |i\rangle\langle j| [/itex]

we obtain a relation true for each \psi,\phi

[itex]\langle\psi|\hat{W}^T|\phi\rangle=\langle\phi|\hat{W}|\psi\rangle [/itex] (1)

Now let's write the transposition super-operator as

[itex]\Lambda[\hat{W}]=\sum_{i,j} |i\rangle\langle j| \hat{W} |i\rangle\langle j| = \hat{W}^T[/itex]

Now seems that the

**transposed matrix**does**not depend**from the**basis**,I have to admit it because the formula (1) is true for each vector and don't depend from the basis.

But that's sound strange to me, because if I have an operator, it can always be diagonal in a certain basis, and so it is equal to its transposition.

We can always write W diagonal in a certain basis [itex]\{|\tilde{k}>\}[/itex] not neccesary equal to [itex]\{|i>\}[/itex]:

[itex]

\hat{W}=\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|[/itex]

and from here do i get

[itex]

\hat{W}^T=(\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|)^T=(\sum_{k} c_k |\tilde{k}\rangle\langle\tilde{k}|=\hat{W}[/itex]

Where i'm wrong?

Can I write every super operator in a representation that is independent from the basis?

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