On Lieb's entropy conjecture

The reformulation of Lieb's entropy conjecture, in the frame of the harmonic analysis on the SO(3) group, makes it evident that the exact value of the classical entropy of a pure quantum state, which belongs to the Hilbert space Hsub(J) of a (2J+1) - dimensional, unitary, irreducible representation Usub(J) of the SO(3) group, depends only on the parameters which characterize the orbits of Usub(J) in Hsub(J). In the case J = 1 we give the exact analytic dependence of the classical entropy of a quantum state on the parameter which characterizes the orbits and as a consequence we obtain a verification of Lieb's entropy conjecture. We verify this conjecture also for any value of J for the states of the canonical basis of Hsub(J). A natural generalization of Lieb's entropy conjecture, which is a new ''phenomenon'' in the harmonic analysis on SO(3), is discussed in the case J = 1. (author)