# Quantum master equation and off-diagonal terms

I have a couple of related questions

1. What is exactly the difference between the quantum master equation and the regular master equation? My understanding is that the normal master equation is used to find a "vector" of state probabilities (like in a regular Markov chain), whereas in the quantum master equation one finds the density matrix. Is this correct? If this is the case, how does the "transition matrix" look, in the quantum case?
2. I'm also a bit confused about the off-diagonal elements in a density matrix. As the density matrix is self-adjoint it can be diagonalizable in some orthonormal basis. So why do we speak about off-diagonal elements? Is it because the density matrix $\rho$ can be time dependent (and orthonormal basis stop being so as time evolves $\Rightarrow$ off-diagonal elements appear)?

The off-diagonal terms appear when you analyze the measurement problem. Say you have a system S with some observable $\hat{S}$ such that it haves eigenvalues $s_i$ and you wish to measure the state. In order to do that you consider an apparatus $\hat{A}$ wich is initially in a pointer state $| a_0 \rangle$ so the initial state of your system is given by the tensor product:

$|\Psi \rangle = |s \rangle \bigotimes |a_0 \rangle$

The act of measurement is the interaction between the apparatus and your system so schematically your Hamiltonian will be:

$H = \hbar \Omega \sigma_z \bigotimes z$

But in general that interaction will form an EPR like state between the system and the apparatus:

$|\Psi \rangle = |s \rangle \bigotimes |a_0 \rangle \mapsto \sum c_i |s_i \rangle |a_i \rangle$

So now your density matrix has off diagonal terms wich in general you don't see. So how do we kill them? Via decoherence from the environment

For the first question on master equation, it turns out that there are loads of equations from different fields that are respectively being called master equations, but they are not related in any certain ways. The Markov Chain master equation and the quantum master equation are one example of this. It seems like you understand the Markov Chain master equation, so I'll just explain the quantum master equation. Simply put, the quantum master equation is a generalization of the Schrodinger's equation to general open quantum systems. For example, if we have a quantum communication channel, where we want to distribute a pair of entangled photons to two different people over a long distance, this channel would obviously be an open system since the photons are coupled to the environment. To address this, we would need to use the quantum master equation.

You can understand the quantum master equation this way. Suppose you consider both the environment and the system you are interested in, then the total system is essentially closed, and you can write down time evolution of the density matrix of the entire system

$$\dot{\rho} = - \frac{i}{\hbar}\left[H_{tot}, \rho\right] = - \frac{i}{\hbar}\left[H_{env} + H_{sys} + H_{coupling}, \rho\right]$$

The quantum master equation is is result of tracing out the environment degrees of freedom and leaving only the system.