I am trying to find the Kraus operator from process matrix.
For instance, suppose that for single qubit identity gate, I have the following process matrix:
[[1. , 0. , 0. , 0. ],
[0. , 0.937, 0.004, 0.005],
[0.023, 0.014, 0.935, 0.03 ],
[0.011, 0.009, 0.017, 0.983]]
From this process matrix, how can I find the Kraus operator(or is it possible to write Kraus operator from process matrix)?
P.S I watched the following steps and I am probably in the wrong way:
1)I started to write ideal process matrix by using definition of super operator. I found the linear map between initial and final density matrix for ideal process matrix As a result, I found:
[[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],
[0., 0., 0., 1.]]
2)Then I started (actually I tried to write) to write the linear map between initial and final density matrix for actual process matrix. And my steps are :
I wrote the kraus operator for single qubit:
$∑k(a∗kI+b∗kX+c∗kY+d∗kZ)ρ(akI+bkX+ckY+dkZ)$
First and third term in this equation are equal and it corresponds to the following matrices:
/
[[a+d, b-ic],
[b+ic, a-d]]
and initial density matrix corresponds:
[[rho00, rho01],
[rho10, rho11]]
- so the final density matrix corresponds:
/
[[(a + d)*(rho00*(a + d) + rho10*(b - 1.0*I*c)) + (b + 1.0*I*c)*(rho01*(a + d) + rho11*(b - 1.0*I*c))
(a - d)*(rho01*(a + d) + rho11*(b - 1.0*I*c)) + (b - 1.0*I*c)*(rho00*(a + d) + rho10*(b - 1.0*I*c))]
[(a + d)*(rho00*(b + 1.0*I*c) + rho10*(a - d)) + (b + 1.0*I*c)*(rho01*(b + 1.0*I*c) + rho11*(a - d))
(a - d)*(rho01*(b + 1.0*I*c) + rho11*(a - d)) + (b - 1.0*I*c)*(rho00*(b + 1.0*I*c) + rho10*(a - d))]]
Now the first row in my ideal and real process matrix has the same values: [1,0,0,0]. This means that my rho00 is actually in the same place. It gives me the following equation:
$rho00 = (a + d)*(rho00*(a + d) + rho10*(b - 1.0*I*c)) + (b + 1.0*I*c)*(rho01*(a + d) + rho11*(b - 1.0*I*c))$
And from this equation, I can say that b and c are equal 0 and a= 1 - d but I can't do anything else and I feel my way is wrong. Although I could find some thing, this is because of process matrix values came likely