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Lieb conjecture

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

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 .

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