How to find the Kraus operators from the process matrix?

Question

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)?

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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

Answer 1

There might be a better way to do this (directly converting to Kraus operators from process matrix), but my suggestion would be to convert from the $\chi$ process matrix to the Choi matrix $\mathcal{C}$ $$ \mathcal{C} = \sum_{i,j}\chi_{mn}|\mathcal{P}_m\rangle\rangle\langle\langle\mathcal{P}_n|, $$ where $|\mathcal{P}_n\rangle\rangle$ is the superket representation of the $N$-qubit Pauli operator $\mathcal{P}_n \in \{I, X, Y, Z \}^{\otimes N}$. We can then use $\mathcal{C}$ to find its eigenvalues $\lambda_i$ and eigenvectors $|M_i\rangle\rangle$, where $|M_i\rangle\rangle$ is the superket representation of the Kraus operators $M_i$.

If you're doing this numerically, there are packages that can do this for you, for example the function chi2kraus in forest-benchmarking.