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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 ρ that is trace-preserving and completely positive for any initial condition.

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

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

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.

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

where the eigenvalues γ_{i} are non-negative. If we define another orthonormal operator basis

we can rewrite Lindblad equation in *diagonal* form

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

and also under the inhomogeneous transformation

However, the first transformation destroys the orthonormality of the operators A_{i} (unless all the γ_{i} are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γ_{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.

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has

Here 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.

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