Lindblad equation

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. 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.

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

where $H$ is a (Hermitian) Hamiltonian part, and $A m$ are an arbitrary orthonormal basis of the operators on the system's Hilbert space with the restriction that $A N 2$ is proportional to the identity operator. Our convention implies that the other $A m$ are traceless. Note that the summation only runs to $N 2 - 1$ . 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 $\{a,b\}=ab+ba.$

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