Abstract. Symplectic tomographies of classical and quantum states are shortly reviewed. The concept of nonlinear f -oscillators and their properties are recalled. The tomographic probability representations of oscillator coherent states and the problem of entanglement are then discussed. The entanglement of even and odd f -coherent states is evaluated by the linear entropy.

Tomographic probability representation of multimode electromagnetic field states in the scheme of center-of-mass tomography is reviewed. Both connection of the field state Wigner function and observable Weyl symbols with the center-of-mass tomograms as well as connection of Grönewold kernel with the center-of-mass tomographic kernel determining the noncommutative product of the tomograms are obtained. The dual center-of-mass tomogram of the photon states are constructed and the dual tomographic kernel is obtained. The models of other generalised center-of-mass tomographies are discussed. Example of two-mode Schrödinger cat states is presented in details.

The problem of quantum particle moving in Dirac delta potential with instant changing well depth is studied by using formalism of tomographic representation of quantum mechanics.The bound state tomogram is given in terms of error function.The ionisation probability due to instant change of the potential parameter is calculated in terms of integral containing the state tomograms .The probability of ionisation is also expressed in terms of the Wigner function.

We review the replica method for calculating the von Neumann entropy and obtain an explicit expression for the entropy of the mixed coherent states |α and |β employing this method. We study the purity inequality for a bipartite system for separable states on the example of even and odd Schrödinger cat state.