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 The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima and Robert Zwanzig) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the ...

, which is in general as difficult to solve as the full quantum problem. The

Redfield equation In quantum mechanics, the Redfield equation is a Markovian master equation that describes the time evolution of the reduced density matrix of a strongly coupled quantum system that is weakly coupled to an environment. The equation is named in ho ...

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). Numerically exact approaches to the kinds of problems to which master equations are usually applied include numerical Feynman integrals, quantum Monte Carlo,

DMRG The density matrix renormalization group (DMRG) is a numerical Variational method (quantum mechanics), variational technique devised to obtain the Macroscopic scale, low-energy physics of Many-body problem, quantum many-body systems with high accu ...

and NRG,

MCTDH Multi-configuration time-dependent Hartree (MCTDH) is a general algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles. MCTDH can thus determine the quantal motion o ...

, and HEOM.

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References