Energy-sensitive and "classical-like" distances between quantum states

We introduce the concept of the "polarized" distance, which distinguishes the orthogonal states with different energies. We also give new inequalities for the known Hilbert-Schmidt distance between neighbouring states and express this distance in terms of the quasiprobability distributions and the normally ordered moments. Besides, we discuss the distance problem in the framework of the recently proposed "classical-like" formulation of quantum mechanics, based on the sympletic tomography scheme. The examples or Fock's, coherent, "Schrodinger cats", squeezed, phase and thermal states an considered.