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

Elliott H. Lieb, Jan Philip Solovej

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


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 (Formula presented.) corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each J and (Formula presented.) 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.

Original languageEnglish (US)
Pages (from-to)379-398
Number of pages20
JournalActa Mathematica
Issue number2
StatePublished - Jun 2014

All Science Journal Classification (ASJC) codes

  • General Mathematics


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