In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad) or master equation in Lindblad form, is the most general type of Markovian and time-homogeneous master equation describing (in general non-unitary) evolution of the density matrix ρ that is trace-preserving and completely positive for any initial condition.[1] The Schrödinger equation is a special case of the more general Lindblad equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation.[2]

## Definition

The Lindblad master equation for an N-dimensional system's density matrix ρ can be written as[1]

${\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{n,m=1}^{N^{2}-1}h_{nm}\left(A_{n}\rho A_{m}^{\dagger }-{\frac {1}{2}}\left\{A_{m}^{\dagger }A_{n},\rho \right\}\right)}$

where H is a (Hermitian) Hamiltonian part, and Am are an arbitrary orthonormal basis of the operators on the system's Hilbert space with the restriction that AN2 is proportional to the identity operator. Our convention implies that the other Am are traceless. Note that the summation only runs to N2 − 1. The coefficient matrix h, together with the Hamiltonian, determines the system dynamics. It must be positive semidefinite to ensure that the equation is trace-preserving and completely positive. The anticommutator is defined as ${\displaystyle \{a,b\}=ab+ba.}$

If the hmn are all zero, then this is quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation.

### Diagonalization

Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u:

${\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}$

where the eigenvalues γi are non-negative. If we define another orthonormal operator basis

${\displaystyle L_{i}=\sum _{j=1}^{N^{2}-1}u_{ji}A_{j}}$

we can rewrite Lindblad equation in diagonal form

${\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i=1}^{N^{2}-1}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right).}$

The new operators Li are commonly called the Lindblad operators of the system. Each term in the sum can be described using the Lindblad superoperator

${\displaystyle D(L)\rho =L\rho L^{\dagger }-{\frac {1}{2}}\left\{L^{\dagger }L,\rho \right\}.}$

## Invariance properties

The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants,

${\displaystyle {\sqrt {\gamma _{i}}}L_{i}\to {\sqrt {\gamma _{i}'}}L_{i}'=\sum _{j=1}^{N^{2}-1}v_{ij}{\sqrt {\gamma _{j}}}L_{j},}$

and also under the inhomogeneous transformation

${\displaystyle L_{i}\to L_{i}'=L_{i}+a_{i}I,}$
${\displaystyle H\to H'=H+{\frac {1}{2i}}\sum _{j=1}^{N^{2}-1}\gamma _{j}\left(a_{j}^{*}L_{j}-a_{j}L_{j}^{\dagger }\right)+bI,}$

where ai are complex numbers and b is a real number. However, the first transformation destroys the orthonormality of the operators Li (unless all the γi are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γi, the Li 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.

## Heisenberg picture

The Lindblad-type evolution of the density matrix in the Schrödinger picture can be equivalently described in the Heisenberg picture using the following (diagonalized) equation of motion for each quantum observable X:

${\displaystyle {\dot {X}}={\frac {i}{\hbar }}[H,X]+\sum _{i=1}^{N^{2}-1}\gamma _{i}\left(L_{i}^{\dagger }XL_{i}-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},X\right\}\right).}$

A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem. Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is unital, i.e. it preserves the identity operator.

## Physical derivation

The Lindblad master equation describes well the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir.[1] Note that the H appearing in the equation is not necessarily equal to the bare system Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction.

## Harmonic oscillator example

The most common Lindblad equation describing the damping of a quantum harmonic oscillator coupled to a reservoir has

{\displaystyle {\begin{aligned}L_{1}&=a,&\gamma _{1}&={\tfrac {\gamma }{2}}\left({\overline {n}}+1\right),\\L_{2}&=a^{\dagger },&\gamma _{2}&={\tfrac {\gamma }{2}}{\overline {n}}.\end{aligned}}}

Here ${\displaystyle {\overline {n}}}$ is the mean number of excitations in the reservoir damping the oscillator and γ 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.