Complete but not overcomplete subset of coherent state Define $$|z \rangle=e^{z \hat b^\dagger-z^* \hat b}|0\rangle$$
$|z\rangle$ is called coherent state, and we know $\{|z \rangle| z\in\mathbb{C}\}$ is overcomplete. So does there exist a complete but not overcomplete subset? How to construct it explicitly?
I try to restrict $z=r e^{i \theta}$ for fixed $r$ or $z\in \mathbb{R}$, but I find they are still overcomplete. 
 A: Complete sets of coherent states can be constructed in many ways. Historically, the most important examples of these complete sets are the Bargmann-Gabor frames, whose completeness was conjegtured by von-Neumann and proved by Perelomov and independently by 
Bargmann, Buerta, Giradello and Klauder.
These proofs are known today as the coherent state density theorem. 
The Bargmann-Gabor frames are sets of coherent states $|z\rangle$ corresponding to the lattice points:
$$z_{mn} = m \omega_1 + n \omega_2$$
($\omega_1$ and $\omega_2$ are not colinear).
The density theorem states that if the lattice unit cell area is greater than $\pi$ the set is not comlete, if the area is smaller than $\pi$ then it is supercomplete (contains many comlete bases) and if the area is exactly equal to $\pi$, then the set is complete and remains complete if exactly one element is removed. In the latter (complete) case, the set of coherent states is called a tight frame. 
The Bargmann-Gabor frames and frames in general found many applications in mathematics (they consist the basis of the theory of wavelets), signal analysis, and physics.
