# MathSciDoc: An Archive for Mathematician ∫

#### Mathematical Physicsmathscidoc:1701.22002

Acta Mathematica, 212, (2), 379-398, 2012.9
Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum$J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to $${K=J+\frac{1}{2}, J+1, ... ,}$$ with $${K=\infty}$$ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each$J$and $${J < K \leqslant \infty}$$ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.
@inproceedings{elliott2012proof,
title={Proof of an entropy conjecture for Bloch coherent spin states and its generalizations},
author={Elliott H. Lieb, and Jan Philip Solovej},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402949851789},
booktitle={Acta Mathematica},
volume={212},
number={2},
pages={379-398},
year={2012},
}
Elliott H. Lieb, and Jan Philip Solovej. Proof of an entropy conjecture for Bloch coherent spin states and its generalizations. 2012. Vol. 212. In Acta Mathematica. pp.379-398. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402949851789.