# How to construct the jump operator in the Lindblad Equation for a 2-qubit system?

I am working on a project for my degree to model a CNOT gate and I'm trying to do this with the Lindblad equation.

$$\dot{\rho} = -\dfrac{i}{\hbar} [H, \rho] + \sum\limits_i \gamma _i (L_i \rho L _i ^{\dagger} - \dfrac{1}{2} \{ L _i ^{\dagger} L_i , \rho \})$$

In the case of modelling a single qubit that might undergo a decay, a quantum jump, finding the jump operator is quite simple. However if we have a combined system of two qubits in a basis $$|00>, |01>, |10>, |11>$$ then it seems to me that the decay is more complicated. For example if it's in state $$|01>$$ then it might decay to $$|00>$$, but if it's in state $$|11>$$ then maybe it will decay to $$|01>$$ or $$|10>$$? And what if it is in a superposition of these states? It seems much more complicated than the single-qubit case where a quantum jump always causes the state to go to $$|0>$$ and nothing else.

I have no idea if this line of thinking is correct, or even makes sense. But I am reaching a bit above my level here in this so forgive me. I suppose what I am asking is what the jump operator(s?) would be in this situation?

One can narrow down the possible options for the Lindblad operators with information about the hierarchy of energy- / time-scales in the system. A typical scenario is that the system-environment interaction energy is much smaller than all other relevant energies. In this case, the Lindblad operators are ladder operators with respect to the Hamiltonian, i.e., $$[H, L_i] = \hbar\omega_i\, L_i$$ for some frequencies $$\omega_i$$. For one qubit with $$H \propto \sigma_z$$, this actually leaves three options: $$L_i \propto \sigma_\pm$$ (spontaneous emission or excitation) or $$L_i \propto \sigma_z$$ (pure dephasing). For two qubits in this scenario, you would need to diagonalize your Hamiltonian $$H$$ to find the eigenstates $$E_k$$, and the Lindblad operators would have the form $$L_i \propto | E_k \rangle\langle E_{k'} |$$.
Another typical scenario is one where the qubit-qubit interaction is even weaker than the qubit-environment interaction. In this case you get what is called a "local Lindblad" equation, where the Lindblad operators are ladder operators for one of the qubits. That is, we get Lindblad operators of the form $$L_i \propto \sigma_{\pm,z}^{(n)}$$ acting only on qubit $$n=1$$ or $$n=2$$. Yet another interesting scenario is one where the qubits do not interact directly, but couple collectively to the same environment. That gives Lindblad operators like $$L = \sum_n \sigma_-^{(n)}$$, leading to the effect of superradiance.
I am not an expert on quantum circuits, but it seems to me that people in this field usually use local decay $$\sigma_-^{(n)}$$ or local dephasing $$\sigma_z^{(n)}$$ Lindblad operators, acting only one one qubit. For example, let's have a look at the qutip-qip software package which allows the simulation of noisy quantum circuits. When creating a "Processor" object, one can directly specify $$T_1$$ and $$T_2$$ times for each qubit, which correspond to local decay and dephasing channels. However, with a bit more effort, one can also specify an arbitrary Lindblad operator by adding "DecoherenceNoise".