Classical analogs of the covariance matrix, purity, linear entropy, and von Neumann entropy

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