6
votes
Prove ${\rm tr}(\rho^2)={\rm tr}((\rho \otimes \rho')S)$ with $S$ the swap operator
In fact, we have a more general equality:
$$
{\text Tr}(AB) = {\text Tr}((A\otimes B) S),
$$
for any square matrices $A,B$.
If $A = |a\rangle\langle b|$ and $B = |c\rangle\langle d|$ then
$$
{\text Tr}...
- 7,612
4
votes
Accepted
What is the operator-sum representation of the two-qubit depolarizing channel?
As pointed out in the comments, you cannot use the one-qubit formula because something like $X\rho X$ does not make sense if $\rho$ is a 2-qubit state. In fact, for this reason the answer you based ...
- 2,823
2
votes
Prove ${\rm tr}(\rho^2)={\rm tr}((\rho \otimes \rho')S)$ with $S$ the swap operator
You can easily prove that identity using graphical notation that depicts tensor contractions by lines:
EDIT: It seems like the graphical notation is less known than I anticipated. Doesn't matter. The ...
- 482
1
vote
Accepted
Writing a Density matrix in terms of the magnitude of the Bloch Vector
Not only is the missing $\frac12$ is a typo, but the squares are a typo, as well. The easiest way to see this (and to verify OP's solution) is to use the eigenvalue formula for $2\times 2$ matrices:
$$...
- 2,823
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