Lindblad master equation (also called the GKSL equation, after Gorini, Kossakowski, Sudarshan, and Lindblad) is the most general Markovian master equation for the time evolution of the density matrix $\rho$ of an open quantum system — one that interacts with its environment.

$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H, \rho] + \sum_k\left(L_k\rho L_k^\dagger - \frac{1}{2}L_k^\dagger L_k\rho - \frac{1}{2}\rho L_k^\dagger L_k\right)$$

The first term $-i[H,\rho]/\hbar$ is the von Neumann term — the coherent Hamiltonian evolution, which is the density matrix form of the Schrödinger equation. The second term is the dissipator, which captures irreversible processes such as decoherence and relaxation due to coupling with the environment. The operators $L_k$ are called Lindblad operators or jump operators and describe the specific noise channels.

Each Lindblad operator $L_k$ describes one noise channel. For a qubit undergoing spontaneous emission (decay from $\lvert 1\rangle$ to $\lvert 0\rangle$) at rate $\gamma$, the jump operator is $L = \sqrt{\gamma}\lvert 0\rangle\langle 1\rvert$. For pure dephasing at rate $\gamma_\phi$, the jump operator is $L = \sqrt{\gamma_\phi/2}\,Z$. Multiple jump operators can be included simultaneously to model multiple noise channels acting in parallel.

The Lindblad equation preserves all physical properties of the density matrix: $\rho$ remains Hermitian, positive semidefinite, and unit trace at all times, ensuring the state is always physically valid. This is guaranteed by the form of the dissipator, which is a completely positive trace-preserving (CPTP) map. In the absence of dissipation ($L_k = 0$ for all $k$), the equation reduces to the closed-system von Neumann equation $d\rho/dt = -i[H,\rho]/\hbar$.