Lieb conjecture explained

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]

External links

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

Notes and References

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