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Wehrl used Glauber coherent states to define a map from quantum densitymatrices to classical phase space densities and conjectured that for Glaubercoherent states the mininimum classical entropy would occur for densitymatrices equal to projectors onto coherent states. This was proved by Lieb in1978 who also extended the conjecture to Bloch SU(2) spin-coherent states forevery angular momentum $J$. This conjecture is proved here. We also recall our1991 extension of the Wehrl map to a quantum channel from $J$ to $K=J+1/2, J+1,...$, with $K=\infty$ corresponding to the Wehrl map to classical densities.For each $J$ and $J<K\leq \infty$ we show that the minimal output entropy forthese channels occurs for a $J$ coherent state. We also show that coherentstates both Glauber and Bloch minimize any concave functional, not justentropy.

Proof of an entropy conjecture for Bloch coherent spin states and its generalizations