  We study the power spectrum of super-Hubble fluctuations of an inflaton-like
scalar field, the "system", coupled to another scalar field, the "environment"
during de Sitter inflation. We obtain the reduced density matrix for the
inflaton fluctuations by integrating out the environmental degrees of freedom.
These are considered to be massless and conformally coupled to gravity as a
\emph{proxy} to describe degrees of freedom that remain sub-Hubble all
throughout inflation. The time evolution of the density matrix is described by
a quantum master equation, which describes the decay of the vacuum state, the
production of particles and correlated pairs and quantum entanglement between
super and sub-Hubble degrees of freedom. The quantum master equation provides a
non-perturbative resummation of secular terms from self-energy (loop)
corrections to the inflaton fluctuations. In the case studied here these are
Sudakov-type double logarithms which result in the \emph{decay} of the power
spectrum of inflaton fluctuations upon horizon crossing with a concomitant
violation of scale invariance. The reduced density matrix and its quantum
master equation furnish a powerful non-perturbative framework to study the
effective field theory of long wavelength fluctuations by tracing short
wavelength degrees of freedom.
