# Lindblad equation
**Lindblad equation** (also called the **GKSL equation**, after Gorini, Kossakowski, Sudarshan, and Lindblad) is the most general Markovian master equation for the [[density-matrix|density matrix]] $\rho$ of an open quantum system. It extends the [[von-neumann-equation|von Neumann equation]] by adding a dissipator term that models incoherent processes — energy loss, dephasing, and other forms of coupling to an environment.

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

## Coherent term

The first term $-\frac{i}{\hbar}[H, \rho]$ is the von Neumann term. It describes the unitary, reversible part of the evolution driven by the system Hamiltonian $H$ — the same dynamics that the [[schrodinger-equation|Schrödinger equation]] captures for pure states. When the dissipator is zero (no environment coupling), the Lindblad equation reduces exactly to the von Neumann equation.

## Dissipator

The sum $\sum_k\!\left(L_k\rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\}\right)$ is the dissipator. Each term corresponds to one incoherent channel indexed by $k$. The $L_k\rho L_k^\dagger$ part is the quantum jump: the environment has "observed" the system through channel $k$, collapsing it toward a new state. The $-\frac{1}{2}\{L_k^\dagger L_k, \rho\}$ part — with $\{A, B\} = AB + BA$ the anticommutator — is the no-jump correction that keeps $\text{tr}(\rho) = 1$ by accounting for the probability that no jump has occurred. Together, the two parts form a completely positive trace-preserving (CPTP) map on $\rho$.

## Jump operators

The operators $L_k$ are called **jump operators** or **Lindblad operators**. Their choice encodes the physics of the environment. Common single-qubit examples:

 - **Amplitude damping** — $L = \sqrt{\gamma}\,\sigma^-$ where $\sigma^- = \lvert 0\rangle\langle 1\rvert$. Models spontaneous emission: the qubit loses energy to the environment at rate $\gamma$ (T1 decay).
 - **Dephasing** — $L = \sqrt{\gamma/2}\,Z$. Models random phase kicks with no energy exchange; destroys off-diagonal coherences at rate $\gamma$ (pure T2 decay).
 - **Depolarizing** — $L_1 = \sqrt{p/3}\,X$, $L_2 = \sqrt{p/3}\,Y$, $L_3 = \sqrt{p/3}\,Z$. Applies each Pauli error with equal probability $p/3$, driving $\rho$ toward the maximally mixed state $I/2$.

## Markov approximation

The Lindblad equation is valid under the **Markov approximation**: the environment is assumed to be memoryless, forgetting the interaction faster than the system evolves. This is analogous to an interrupt-driven peripheral that completes its transaction well within one main-loop cycle — the system never needs to track the peripheral's internal state. When the bath correlation time $\tau_{\text{bath}}$ is comparable to the system timescale, memory effects matter and the full non-Markovian dynamics must be used instead.

## List of code implementations

 - [[lindblad-equation-qutip|Lindblad equation (QuTiP)]]