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

TL;DR

This paper proves a conjecture that coherent states minimize classical entropy for certain quantum states, extending previous results to Bloch spin states and related quantum channels, and showing they minimize all concave functionals.

Contribution

The paper proves Wehrl's entropy conjecture for Bloch SU(2) spin-coherent states and extends the minimal entropy results to associated quantum channels and concave functionals.

Findings

Proved Wehrl's entropy conjecture for Bloch coherent states.

Established minimal output entropy for quantum channels at coherent states.

Demonstrated that coherent states minimize all concave functionals.

Abstract

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 $K=J+1/2, J+1, ...$, 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 for these channels occurs for a $J$ coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.