  We obtain a classical analog of the quantum covariance matrix by performing
its classical approximation for any continuous quantum state, and we illustrate
this approach with the anharmonic oscillator. Using this classical covariance
matrix, we propose classical analogs of the purity, linear quantum entropy, and
von Neumann entropy for classical integrable systems, when the quantum
counterpart of the system under consideration is in a Gaussian state. As is
well known, this matrix completely characterizes the purity, linear quantum
entropy, and von Neumann entropy for Gaussian states. These classical analogs
can be interpreted as quantities that reveal how much information from the
complete system remains in the considered subsystem. To illustrate our
approach, we calculate these classical analogs for three coupled harmonic
oscillators and two linearly coupled oscillators. We find that they exactly
reproduce the results of their quantum counterparts. In this sense, it is
remarkable that we can calculate these quantities from the classical viewpoint.
