Optimal Concentration for SU(1,1) Coherent State Transforms and an analogue of the Lieb-Wehrl Conjecture for SU(1,1)

TL;DR

This paper establishes a lower bound for Wehrl entropy in SU(1,1) coherent states, matching the Lieb-Wehrl conjecture at high quantum numbers, using sharp Sobolev inequalities and a novel uniqueness theorem on the hyperbolic plane.

Contribution

It introduces a new family of sharp Sobolev inequalities on the hyperbolic plane and proves a uniqueness theorem for semi-linear Poisson equations, advancing the understanding of entropy bounds in SU(1,1) quantum states.

Findings

Lower bound for Wehrl entropy in SU(1,1) matches the conjecture at high quantum numbers.

Development of a new sharp Sobolev inequality on the hyperbolic plane.

Proved a novel uniqueness theorem for semi-linear Poisson equations on hyperbolic geometry.

Abstract

We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1,1) coherent state transforms and a new family of sharp Sobolev inequalities on the hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on the hyperbolic plane. Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities.