Lieb conjecture

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,[1] who at the same time extended it to the SU(2) case. The conjecture was proven in 2012, by Lieb and Jan Philip Solovej.[2] The uniqueness of the minimizers was only proved in 2022 by Rupert L. Frank[3] and Aleksei Kulikov, Fabio Nicola, Joaquim Ortega-Cerda' and Paolo Tilli.[4]

References

1. Lieb, Elliott H. (August 1978). "Proof of an entropy conjecture of Wehrl". Communications in Mathematical Physics. 62 (1): 35–41. Bibcode:1978CMaPh..62...35L. doi:10.1007/BF01940328. S2CID 189836756.
2. Lieb, Elliott H.; Solovej, Jan Philip (17 May 2014). "Proof of an entropy conjecture for Bloch coherent spin states and its generalizations". Acta Mathematica. 212 (2): 379–398. arXiv:1208.3632. doi:10.1007/s11511-014-0113-6. S2CID 119166106.
3. Frank, Rupert L. (2023). "Sharp inequalities for coherent states and their optimizers". Advanced Nonlinear Studies. 23 (1): Paper No. 20220050, 28. arXiv:2210.14798. doi:10.1515/ans-2022-0050.
4. Kulikov, Aleksei; Nicola, Fabio; Ortega-Cerda', Joaquim; Tilli, Paolo (2022). "A monotonicity theorem for subharmonic functions on manifolds". arXiv:2212.14008 [math.CA].

External links

• Video of a lecture by Lieb discussing the conjecture and outlining its proof.