Quantum master equation and off-diagonal terms

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