Lieb conjecture

{{short description|Theorem in quantum information theory}}

In quantum information theory, the Lieb conjecture is a theorem concerning the Wehrl entropy of quantum systems for which the classical phase space is a sphere. It states that no state of such a system has a lower Wehrl entropy than the SU(2) coherent states.

The analogous property for quantum systems for which the classical phase space is a plane was conjectured by Alfred Wehrl in 1978 and proven soon afterwards by Elliott H. Lieb,{{cite journal|last1=Lieb|first1=Elliott H.|title=Proof of an entropy conjecture of Wehrl|journal=Communications in Mathematical Physics|date=August 1978|volume=62|issue=1|pages=35–41|doi=10.1007/BF01940328|bibcode=1978CMaPh..62...35L|s2cid=189836756|url=http://projecteuclid.org/euclid.cmp/1103904300}} who at the same time extended it to the SU(2) case.

The conjecture was proven in 2012, by Lieb and Jan Philip Solovej.{{cite journal|last1=Lieb|first1=Elliott H.|last2=Solovej|first2=Jan Philip|title=Proof of an entropy conjecture for Bloch coherent spin states and its generalizations|journal=Acta Mathematica|date=17 May 2014|volume=212|issue=2|pages=379–398|doi=10.1007/s11511-014-0113-6|arxiv=1208.3632|s2cid=119166106}} The uniqueness of the minimizers was only proved in 2022 by Rupert L. Frank{{cite journal |last1=Frank |first1=Rupert L. |title=Sharp inequalities for coherent states and their optimizers |journal=Advanced Nonlinear Studies |date=2023 |volume=23 |issue=1 |page=Paper No. 20220050, 28 |doi=10.1515/ans-2022-0050 |url=https://doi.org/10.1515/ans-2022-0050|arxiv=2210.14798 }} and Aleksei Kulikov, Fabio Nicola, Joaquim Ortega-Cerda' and Paolo Tilli.{{cite arXiv |last1=Kulikov |first1=Aleksei |last2=Nicola |first2=Fabio |last3=Ortega-Cerda' |first3=Joaquim |last4=Tilli |first4=Paolo |title=A monotonicity theorem for subharmonic functions on manifolds |date=2022 |class=math.CA |eprint=2212.14008}}