  We study the quantum to classical transition in a chaotic system surrounded
by a diffusive environment. The emergence of classicality is monitored by the
Renyi entropy, a measure of the entanglement of a system with its environment.
We show that the Renyi entropy has a transition from quantum to classical
behavior that scales with $\hbar^2_{\rm eff}/D$, where $\hbar_{\rm eff}$ is the
effective Planck constant and $D$ is the strength of the noise. However, it was
recently shown that a different scaling law controls the quantum to classical
transition when it is measured comparing the corresponding phase space
distributions. We discuss here the meaning of both scalings in the precise
definition of a frontier between the classical and quantum behavior. We also
show that there are quantum coherences that the Renyi entropy is unable to
detect which questions its use in the studies of decoherence.
