Proof of an Entropy Conjecture of Wehrl

Élliott H. Lieb

Shunlong Luo

Based on the hypercontractivity of the free harmonic oscillator Hamiltonian and the canonical commutation relation, we present a simple proof of Wehrl's conjecture on the Shannon entropy of phase space probability densities in the Fock-Bargmann representation.

Ayumu Sugita

Gnutzmann and Życzkowski have proposed the Rényi–Wehrl entropy as a generalization of the Wehrl entropy, and conjectured that its minimum is obtained for coherent states. We prove this conjecture for the Rényi index q = 2, 3, ... in the cases of compact semisimple Lie groups. A general formula for the minimum value is given.

Kedar S. Ranade, Wolfgang P. Schleich

Sven Gnutzmann∥, Karol Życzkowski

We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.

Alfred Wehrl