Title: Optimal Concentration for SU(1,1) Coherent State Transforms and
an Analogue of the Lieb-Wehrl Conjecture for SU(1,1)
Abstract:
The coherent state transform is an isometry from the quantum state
space of a system into the Hilbert space of square-integrable functions
on the corresponding classical phase space. Wehrl proposed defining the
classical entropy of a quantum state via its coherent state transform
and conjectured a lower bound on this entropy. Specifically, he
conjectured that when the classical phase space is the Euclidean plane,
the lower bound is attained by Glauber coherent states. This conjecture
was then proved by Lieb, who extended the conjecture to SU(2) coherent
state transforms, for which the classical phase space is the
two-dimensional sphere. This conjecture is still open, although a nice
asymptotic result was obtained by Bodmann recently.
Here I derive a lower bound for the Wehrl entropy in the setting of
SU(1,1), for which the classical phase space is the two-dimensional
hyperbolic plane. 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 on
the hyperbolic plane and a new family of sharp Sobolev inequalities on
the hyperbolic plane. To prove the sharpness of the Sobolev inequality,
one needs 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. In the first part of the talk, I
shall review Wehrl's original conjecture and Lieb's proof and extension
of it. The second part of the talk will concern the new results proved
for SU(1,1).