This paper proves a clear, general statement: for any compact connected semisimple Lie group, the quantum states built from highest-weight vectors — the coherent projectors — give the smallest possible Wehrl entropy. In plain terms, if you take any finite-dimensional irreducible representation of such a group and turn a quantum state into a probability distribution on the group by measuring its overlap with the group’s coherent states (the Husimi function), the Shannon entropy of that distribution is lowest exactly when the quantum state is one of those coherent states.

The authors set the problem up as follows. Fix a compact connected semisimple Lie group G and an irreducible unitary representation labeled by a highest weight λ. Pick the unit highest-weight vector and let the group move that vector to make a family of rank-one projectors called coherent projectors. For any density matrix ρ (a general mixed quantum state) one defines the Husimi function Qρ(g)=tr(ρP_g), a probability density on the group. The Wehrl-type entropy is the integral −∫Qρ(g) log Qρ(g) dg. The main theorems show that this entropy is minimized precisely when ρ is a coherent projector, and that the same coherent projectors uniquely maximize the Husimi power moments for every order p>1.

At a high level the proof examines what happens if you start with a candidate optimizer and make small perturbations. The perturbations are chosen along directions coming from the Lie algebra — the Killing fields — which represent infinitesimal group motions. Using second-derivative (second-variation) calculations and standard identities for Casimir operators, the problem reduces to a sharp bound on a certain moment map associated with the state. That “extreme-moment” bound says the moment is maximal only for coherent projectors, and combining these facts forces any optimizer to be a coherent state. The method follows the spirit of an earlier strategy by Frank and Lieb for related inequalities on the Heisenberg group.