  It has recently been proposed classical analogs of the purity, linear quantum
entropy, and von Neumann entropy for classical integrable systems, when the
corresponding quantum system is in a Gaussian state. We generalized these
results by providing classical analogs of the generalized purities,
Bastiaans-Tsallis entropies, R\'enyi entropies, and logarithmic negativity for
classical integrable systems. These classical analogs are entirely
characterized by the classical covariance matrix. We compute these classical
analogs exactly in the cases of linearly coupled harmonic oscillators, a
generalized harmonic oscillator chain, and a one-dimensional circular lattice
of oscillators. In all of these systems, the classical analogs reproduce the
results of their quantum counterparts whenever the system is in a Gaussian
state. In this context, our results show that quantum information of Gaussian
states can be reproduced by classical information.
