On Lieb's conjecture

Abstract

The reformulation of Lieb's conjecture, in the frame of the harmonuic 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 of a (2J+1)-dimensional, unitary, irreducible representation of the SO(3) group, depends only on the parameters which characterize the orbits of this representation. In the case J=1 we give the exact analytic dependence of the classical entropy of a quantum state on the parameters which characterize 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 the Hilbert space of the representation. 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.