The decoherence process has allowed us to explain various (classical and decoherence) sources of measurement noise in quantum systems. I intuitively understand this physical concept of decoherence-inducing-noise.
I am trying to understand this concept from an information theoretic perspective. We know that the environment-induced-decoherence takes away state information from a system. That would imply reduction in entropy of the system. Now analyzing the modeling process, the measurement process of a (environment induced) decohering system can be modeled by adding "noise" terms. That would imply addition of entropy to the system. Which is contradicting.
So my question is, from an information perspective, how can the (environment-induced-)decoherence process consistently (theoretically) describe the occurrence of (non-classical) noise? Or, what is the nature of such a noise?
Edit This is an explanation on why I think that decoherence might lead to reduction in entropy. I am using an information theoretic interpretation on entropy for this purpose, which can be interpreted as the amount of information that you lack about the exact physical state of the system (see https://physics.stackexchange.com/q/119515).
The measurement problem has 3 parts (Schlosshauer book on Decoherence). For this discussion only first 2 are relevant: 1) the problem of preferred basis and 2) the problem of non observability of interference.
Consider an unknown system that is initially isolated. Now I (or the environment) make measurements on this system and that makes the system vulnerable to decoherence. So it happens that through einselection I can find a set of robust states that are least prone to decoherence (pointer states) Let's call them $|\Psi_1\rangle$ and $|\Psi_2\rangle$. So this process allowed me to gain more information about the system, then I had before. Hence, the reduction in entropy! Now of course, initially the system was in a pure superposition of these states say $|\Psi\rangle = \alpha |\Psi_1\rangle + \beta |\Psi_2\rangle$, with a definite $\alpha$, $\beta$, therefore zero entropy. But as an observer with no initial knowledge of the system (an alien observer on system earth!) I model my belief on the system as an arbitrary superposition, say $|\Phi\rangle = a|\Phi_1\rangle + b|\Phi_2\rangle$ with unknown $a$, $b$, therefore a positive entropy. Now ofcourse, through the measurement process (and thanks to the physical process of einselection) I can find relationships between $(\alpha, \beta, \Psi_1, \Psi_2)$ and $(a,b,\Phi_1,\Phi_2)$. Hence, this entire process of measurement has 2 competing entropy elements, 1) First entropy element that is reduced by gaining knowledge on the pointer states, hence the basis and coefficients, 2) second entropy element that increases due to the maping of pure states to mixed states.