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 $\rho$ of an open quantum system. It extends the von Neumann equation by adding a dissipator term that models incoherent processes — energy loss, dephasing, and other forms of coupling to an environment.

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

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 Schrödinger equation captures for pure states. When the dissipator is zero (no environment coupling), the Lindblad equation reduces exactly to the von Neumann equation.

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 $\rho$.

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 $I/2$.

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.

• Lindblad equation (QuTiP)