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I'm teaching myself open quantum systems and the concept of a stochastic master equation has arisen. As someone who has studied classical stochastic processes a fair bit, this seems, at least to my naive intuition, like a completely nonsensical quantity. Can someone explain the purpose/functioning of such a thing?

Specifically, my intuition is as follows:

For classical system one has:

Classical quantity subject to uncertain dynamics $\rightarrow$ Noisy/random state (modelled by SDEs for example) $\rightarrow$ Deterministic probability dynamics (Fokker planck or master equation description)

Removing the noise in the SDE means removing higher order terms in the deterministic FP-eqn leaving us with something like the Liouville eqn.

For quantum systems:

A quantum state description subject to uncertain dynamics $\rightarrow$ e.g. a Lindblad equation in the reduced density matrix (which could be unravelled into a stochastic schrodinger equation)

Removing the noise in the Schrodinger eqn removes the (deterministic) non-coherent terms in the Linblad eqn leaving us with the Quantum Liouville eqn.

So we have

Stochastic Dif. eq./jump process $\leftrightarrow$ Deterministic Fokker Planck/Master equation $\rightarrow_{noise free}$ Liouville equation

Stochastic Schrodinger equation $\leftrightarrow$ Deterministic Lindblad equation $\rightarrow_{noise free}$ Quantum Liouville equation

I.e. the Lindblad equation, describing the evolution of the ensemble of states under uncertain dynamics, is deterministic. What then, should I make of things like this (https://en.wikipedia.org/wiki/Belavkin_equation) where there are additional stochastic terms in the Lindblad equation? They claim to average to the Lindblad equation.

But this seems to be getting the levels of description/sources of uncertainty extremely muddled: no one adds stochastic terms to a classical master equation (or FP eqn), the entire point is that they describe the ensemble behaviour that results from the underlying stochastic dynamics.

It implies one would need/could build a 'super-master equation' which would be deterministic in the dynamics of distributions of (apparently stochastic) density matrices...

But density matrices are perfectly capable of describing mixed states (indeed that's the point), so isn't the above just pathological and unnecessary?

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