# Understanding quantum stochastic master equations

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?

The Belavkin equation and other stochastic master equations describe the evolution of a system which is being continuously measured. Since quantum measurement is inherently stochastic, this is what leads to the 'extra' layer of stochasticity that you have noticed.

As you correctly noted, the Lindblad equation is deterministic, as it represents an 'averaging out' of the system-environment interaction. Mathematically, this is achieved by performing a partial trace over the environment following a unitary system-environment interaction. There is no measurement involved in the dynamics that lead to Linblad equation.

Stochastic master equations (SME), on the other hand, do involve continuous measurements on some part of the system. The outcomes of these measurements are stochastic, so the exact dynamics will change depending on what outcomes you get. For example, the SME for a continuously measured continuous variable $$X$$ is:

$$d \rho = -k[X,[X,\rho]]dt + \sqrt{2k} (X \rho + \rho X - 2 \langle X\rangle \rho)dW$$ where $$\rho$$ is the density operator of the system, $$k$$ is a determines the strength of the measurement and $$dW$$ is a Wiener Process-a continuous stochastic process. If you evolve a state using this equation multiple times, starting from the same state each time, you will find that the dynamics each time will be different, due to the inherent randomness of the quantum measurement process captured by $$dW$$. These different evolutions are sometimes known as 'unravellings' of the master equation.

It is important to note that in the above equation, the density operator acts as a'state of knowledge' of the observer. In other words, it assumes that the observer is keeping track of the measurement result. If the measurement result is discarded, then one ends up with deterministic evolution of the density operator, as the stochastic nature of the measurement is 'averaged out'. Mathematically, this is achieved by setting the the term proportional to $$dW$$ equal to 0 and retaining only the term proportional to $$dt$$.

For a derivation of the above equation and further discussion, I recommend 'A Straightforward Introduction of Continuous Quantum Measurement' by Jacobs and Steck.