Quantum information

The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.^a Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.

Given an arbitrary orthonormal basis $L_n$ of the space of operators on the $N$ -dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product $\langle A,B \rangle = \mathrm{Tr}[A^\dagger B]$ ), the Lindblad equation takes the following form:

(1) $\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\left(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\right) , \end{align*}$