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| lindbald-equation [May 25, 2026 at 13:52] – created - external edit 127.0.0.1 | lindbald-equation [June 09, 2026 at 22:57] (current) – removed Ivan Janevski |
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| # Lindblad master equation | |
| **Lindblad master equation** (also called the GKSL equation, after Gorini, Kossakowski, Sudarshan, and Lindblad) is the most general Markovian master equation for the time evolution of the density matrix $\rho$ of an open quantum system — one that interacts with its environment. | |
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| $$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H, \rho] + \sum_k\left(L_k\rho L_k^\dagger - \frac{1}{2}L_k^\dagger L_k\rho - \frac{1}{2}\rho L_k^\dagger L_k\right)$$ | |
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| The first term $-i[H,\rho]/\hbar$ is the von Neumann term — the coherent Hamiltonian evolution, which is the density matrix form of the Schrödinger equation. The second term is the **dissipator**, which captures irreversible processes such as decoherence and relaxation due to coupling with the environment. The operators $L_k$ are called **Lindblad operators** or **jump operators** and describe the specific noise channels. | |
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| ## Lindblad operators | |
| Each Lindblad operator $L_k$ describes one noise channel. For a qubit undergoing spontaneous emission (decay from $\lvert 1\rangle$ to $\lvert 0\rangle$) at rate $\gamma$, the jump operator is $L = \sqrt{\gamma}\lvert 0\rangle\langle 1\rvert$. For pure dephasing at rate $\gamma_\phi$, the jump operator is $L = \sqrt{\gamma_\phi/2}\,Z$. Multiple jump operators can be included simultaneously to model multiple noise channels acting in parallel. | |
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| ## Properties | |
| The Lindblad equation preserves all physical properties of the density matrix: $\rho$ remains Hermitian, positive semidefinite, and unit trace at all times, ensuring the state is always physically valid. This is guaranteed by the form of the dissipator, which is a completely positive trace-preserving (CPTP) map. In the absence of dissipation ($L_k = 0$ for all $k$), the equation reduces to the closed-system von Neumann equation $d\rho/dt = -i[H,\rho]/\hbar$. | |
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