# From Lindblad operators to Kraus operators: an explicit example of a dephasing noise model

I'm trying to understand how to obtain a set of Kraus operators from Lindblad master equations. For a $$1$$-qubit dephasing noise model, it is well-known that the set of Kraus operators is $$\{ \sqrt{p}I, \sqrt{1-p}Z \}$$ and Lindblad operator is $$L = \sqrt{\gamma} Z$$, where one can express $$p$$ in terms of $$\gamma$$. In fact, this reference provides a great explanation on it.

Now I want to extend this to multiple qubits. For $$2$$-qubits system subjected to a pure local dephasing noise model, Lindblad operators are $$L_i = \sqrt{\gamma}Z_i$$ where $$Z_i$$ is a Pauli-Z operator acting on $$i$$-th qubit (see e.g., this post). What is a set of Kraus operators for this model, and how one should derive it?

The solution of the Lindblad equation is a quantum map $$\Lambda_t$$ such that $$\rho_t = \Lambda_t \rho_0$$. (If the Lindblad equation is time-homogeneous, i.e., $$\dot\rho_t = L \rho_t$$ with constant $$L$$, then $$\Lambda_t = \exp(Lt)$$.) The Kraus operators can be found from the Choi representation of $$\Lambda_t$$ as follows.
Let $$|+\rangle = \frac{1}{\sqrt d} \sum_n |n\rangle \otimes |n\rangle$$ be the maximally entangled state on $$\mathcal H \otimes \mathcal H$$, where $$\mathcal H$$ is the Hilbert space of the system and $$d$$ its dimension. The Choi representation of $$\Lambda$$ ($$t$$ subscripts will be omitted from now on) is then $$J(\Lambda) = \bigl( 1 \otimes \Lambda \bigr)\bigl(\, |+\rangle\langle+|\, \bigr) .$$ Its eigendecomposition can be brought into the form (see for example Norbert Schuch's answer here for details) $$J(\Lambda) = \sum_i (1 \otimes A_i)\, |+\rangle\langle+|\, (1 \otimes A_i)^\dagger$$ and the $$A_i$$ here are the Kraus operators, $$\rho_t = \sum A_i \rho_0 A_i^\dagger$$.
In your concrete problem, if I understand it correctly, you have two subspaces $$\mathcal H_1$$ and $$\mathcal H_2$$, and quantum maps $$\Lambda_1$$ and $$\Lambda_2$$ acting independently on the subspaces. If $$\{ A_i \}_i$$ is the set of Kraus operators for $$\Lambda_1$$ and $$\{ B_j \}_j$$ the one for $$\Lambda_2$$, then $$\{ A_i \otimes B_j \}_{i,j}$$ are Kraus operators for $$\Lambda_1 \otimes \Lambda_2$$.