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 Bell-type state $$b_k \in H_k\otimes H_k$$ . When is compact or when $${\mathbb {C}}^n$$ provide positive lower bounds for entanglement entropy $$b_k$$ (asymptotic k, as $$k\rightarrow \infty $$ ).