Quantum mechanics as the quadratic Taylor approximation of classical mechanics: the finite dimensional case

We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties related to the infinite dimension of phase space for quantum mechanics, we consider in this note the finite dimensional quantum mechanics. On one hand, this is a simple example of the high pedagogical value. On the other hand, quantum information operates in the finite dimensional state space. Therefore our investigation can be considered as construction of a classical statistical model for quantum information.