# Is there a straightforward method to find the Kraus operators of a given quantum channel?

$\newcommand{\Ket}{\left|#1\right>}$ 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:

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

• 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 $E_k$? – Georg Jun 28 '18 at 15:59
• 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). – Norbert Schuch Jun 30 '18 at 13:26
• Thank you for your comment! I've edited the post I hope my question is clearer now. – Alexia Jun 30 '18 at 16:10
• @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). – Norbert Schuch Jun 30 '18 at 17:06