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A **quantum master equation** is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.

A formally exact quantum master equation is the Nakajima–Zwanzig equation, which is in general as difficult to solve as the full quantum problem.

The Redfield equation and Lindblad equation are examples of approximate Markovian quantum master equations. These equations are very easy to solve, but are not generally accurate.

Some modern approximations based on quantum master equations, which show better agreement with exact numerical calculations in some cases, include the polaron transformed quantum master equation and the VPQME (variational polaron transformed quantum master equation).^{[1]}

Numerically exact approaches to the kinds of problems to which master equations are usually applied include numerical Feynman integrals,^{[2]} quantum Monte Carlo, DMRG^{[3]} and NRG, MCTDH,^{[4]} and HEOM.

**^**D. McCutcheon, N. S. Dattani, E. Gauger, B. Lovett, A. Nazir (25 August 2011). "A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots".*Physical Review B*.**84**(8): 081305R. arXiv:1105.6015. Bibcode:2011PhRvB..84h1305M. doi:10.1103/PhysRevB.84.081305. S2CID 119275166.`{{cite journal}}`

: CS1 maint: multiple names: authors list (link)**^**Dattani, Nike (2013), "FeynDyn: A MATLAB program for fast numerical Feynman integral calculations for open quantum system dynamics on GPUs",*Computer Physics Communications*,**184**(12): 2828–2833, arXiv:1205.6872, Bibcode:2013CoPhC.184.2828D, doi:10.1016/j.cpc.2013.07.001, S2CID 41378038**^**Prior, Javier (30 July 2010). "Efficient Simulation of Strong System-Environment Interactions".*Phys. Rev. Lett*.**105**(5): 050404. arXiv:1003.5503. Bibcode:2010PhRvL.105e0404P. doi:10.1103/PhysRevLett.105.050404. PMID 20867899. S2CID 27896369. Retrieved 2 June 2021.**^**Wang, Haobin (24 March 2017). "A multilayer multiconfiguration time-dependent Hartree simulation of the reaction-coordinate spin-boson model employing an interaction picture".*J. Chem. Phys*.**146**(12): 124112. Bibcode:2017JChPh.146l4112W. doi:10.1063/1.4978901. OSTI 1497842. PMID 28388113. Retrieved 2 June 2021.