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$\newcommand{\Ket}[1]{\left|#1\right>}$ I would like to know if there is a systematic way of finding a set of Kraus operators $E_k$ for a quantum channel $\varepsilon$ defined by its action on a density matrix $\rho$ using these properties:

$\varepsilon(\rho) = \sum_{k}E_k\rho E_k ^{\dagger}$

$\sum_{k}E_k^{\dagger} E_k = \mathbb{1}$

I feel like you can "educated" guess the answer but I would like to know if there is a more formal method.

To be more specific, there are 2 different input possibilities for the problem: the first is when the channel $\varepsilon$ is defined by its action on a density matrix $\rho$, and the second is when it is defined by its action on kets.

Let me give an example for both these cases:

  1. Action on $\rho$: find the Kraus operators for the dephasing channel: $\rho \rightarrow \rho ' = (1-p)\rho + p\, diag(\rho_{00},\rho_{01})$

  2. Action on kets: find the Kraus operators for the amplitude damping channel, defined by the action: $\Ket{00} \rightarrow \Ket{00}$, $\Ket{10} \rightarrow \sqrt{1-p}\Ket{10} + \sqrt{p}\Ket{01}$

I cannot figure out a method for any of these types of cases even though I know a possibility of Kraus operators for both of these cases. For the dephasing channel:

$E_0 = \sqrt{1-p/2}\mathbb{1}$ and $E_1 = \sqrt{p/2}\sigma_z$

and for the amplitude damping channel:

$E_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-p} \end{pmatrix} \quad E_1= \begin{pmatrix} 0 & \sqrt{p} \\ 0 & 0 \end{pmatrix}$

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    $\begingroup$ So what is the 'input' to the problem? Are you given the action of the channel on all the pure states and you want to figure out the $E_k$? $\endgroup$ – Georg Jun 28 '18 at 15:59
  • $\begingroup$ I'll second that. You have to specify how you are given the channel, otherwise the question is near impossible to answer (unless one enumerates all possibilities which come to mind). $\endgroup$ – Norbert Schuch Jun 30 '18 at 13:26
  • $\begingroup$ Thank you for your comment! I've edited the post I hope my question is clearer now. $\endgroup$ – Alexia Jun 30 '18 at 16:10
  • $\begingroup$ @Alexia Your "action on kets" notation is unclear to me. Why is this a 2-qubit ket? --- On a more general footing: Would a valid input be in either case one which would allow you to compute the action of the channel on any input state (including part of a larger entangled state)? Then one could base the answer on that premise (which would allow for a rather straightforward answer). In this case I would suggest rephrasing the question like that (and giving those 2 cases as examples). P.S.: You can use @[username] to notify one user above of your comment (otherwise it might go unnoticed). $\endgroup$ – Norbert Schuch Jun 30 '18 at 17:06
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you can use the choi isomorphisme:

you apply the Choi map to your channel to obtain the corresponding Choi matrix and then you compute the spectral decomposition of this matrix. The Kraus operators will be the eigenvectors rearranged (vectorization) into a matrix and the weight of each Kraus operator will be the corresponding eigenvalue.

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Evaluate the partial trace over the environment to get the krauss operators

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