Lindblad Generator

Generator

The Lindblad generator acts on density operators:

$$\frac{d\rho}{dt} = \mathcal L(\rho).$$

In Gorini–Kossakowski–Sudarshan–Lindblad form,

$$\mathcal L(\rho) = -\frac{i}{\hbar}[H,\rho] + \sum_\mu \left( L_\mu\rho L_\mu^\dagger - \frac12 \{L_\mu^\dagger L_\mu,\rho\} \right).$$

Assumptions

• The dynamics is Markovian and generates a completely positive trace-preserving semigroup.
• $H$ is the Hamiltonian part of the generator.
• $L_\mu$ are Lindblad, jump, or noise operators.
• Time-dependent or non-Markovian variants require extra assumptions.

Common Mistakes

• Calling $\mathcal L$ a Hamiltonian; it is a superoperator on density operators.
• Assuming every open-system equation is Lindblad form.
• Forgetting complete positivity when fitting rates.
• Treating Lindblad operators as unique.

Canonical Links

• Full Lindblad–GKSL treatment
• Lindblad equation formula card
• Liouvillian Superoperators
• Kraus Map
• Math Needed for Open Systems
• Density Operator

References

• G. Lindblad, “On the generators of quantum dynamical semigroups”, Communications in Mathematical Physics 48, 119-130, 1976.
• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems”, Journal of Mathematical Physics 17, 821-825, 1976.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.