In quantum mechanics, Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in Lindblad form is the most general type of Markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix $\rho$ that is trace-preserving and completely positive for any initial condition.

Lindblad master equation for an $N$-dimensional system's reduced density matrix $\ \rho$ can be written:

$\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\big(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\big)$

where $\ H$ is a (Hermitian) Hamiltonian part, the $\ L_m$ are an arbitrary orthonormal basis of the operators on the system's Hilbert space, and the $\ h_{n,m}$ are constants which determine the dynamics. The coefficient matrix $\ h = (h_{n,m})$ must be positive to ensure that the equation is trace-preserving and completely positive. The summation only runs to $\ N^2-1$ because we have taken $\ L_{N^2}$ to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the $\ L_m$ are traceless for $\ m. The terms in the summation where $m=n$ can be described in terms of the Lindblad superoperator, $L(C)\rho=C\rho C^\dagger -\frac{1}{2}\left(C^\dagger C \rho +\rho C^\dagger C\right)$.

If the $\ L_m$ terms are all zero, then this is quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

Note that $\ H$ is not necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction.

## Diagonalization

Since the matrix $\ h = (h_{n,m})$ is positive, it can be diagonalized with a unitary transformation u:

$u^\dagger h u = \begin{bmatrix} \gamma_1 & 0 & \cdots & 0 \\ 0 & \gamma_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \gamma_{N^2-1} \end{bmatrix}$

where the eigenvalues $\ \gamma_i$ are non-negative. If we define another orthonormal operator basis

$A_i = \sum_{j = 1}^{N^2-1} u_{j,i} L_j$

we can rewrite Lindblad equation in diagonal form

$\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{i = 1}^{N^2-1} \gamma_{i}\big(A_i\rho A_i^\dagger -\frac{1}{2} \rho A_i^\dagger A_i -\frac{1}{2} A_i^\dagger A_i \rho \big) .$

This equation is invariant under a unitary transformation of Lindblad operators and constants,

$\sqrt{\gamma_i} A_i \to \sqrt{\gamma_i'} A_i' = \sum_{j = 1}^{N^2-1} v_{j,i} \sqrt{\delta_i} A_j ,$

and also under the inhomogenous transformation

$A_i \to A_i' = A_i + a_i ,$
$H \to H' = H + \frac{1}{2i} \sum_{j=1}^{N^2-1} \gamma_j (a_j^* A_j - a_j A_J^\dagger) .$

However, the first transformation destroys the orthonormality of the operators $\ A_i$ (unless all the $\ \gamma_i$ are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the $\ \gamma_i$, the $\ A_i$ of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.

## Harmonic oscillator example

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has $\ L_0=a$, $\ L_1=a^{\dagger}$, $\ h_{0,1}=-(\gamma/2)(\bar n+1)$, $\ h_{1,0}=-(\gamma/2)\bar n$ with all others $\ h_{n,m}=0$. Here $\bar n$ is the mean number of excitations in the reservoir damping the oscillator and $\ \gamma$ is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.