Coherent states and entropy

Let $H_k$, $k\in \mathbb{N}$, be the Hilbert spaces of geometric quantization on a Kähler manifold $M$. With two points in $M$ we associate a Bell-type state $b_k\in H_k\otimes H_k$. When $M$ is compact or when $M$ is ${\mathbb{C}}^n$, we provide positive lower bounds for the entanglement entropy of $b_k$ (asymptotic in $k$, as $k\to \mathrm{\infty }$).