Von Neumann entropy of mixtures of coherent states I'm trying to calculate the Von Neumann entropy of statistical mixtures of coherent states. The problem is that such states are in general non-Gaussian, so one cannot follow the formalism developed here: Phys. Rev. A 59, 1820 (1999). Does anybody have any hints on how to calculate the $$Tr[\rho\log[\rho]],$$ for $$\rho = \sum_i p_i |\alpha_i\rangle\langle\alpha_i|~?$$
 A: It seems I have figured out an answer for 2 terms in the original state.
Suppose that the state is 
$$\rho = a |\alpha \rangle \langle \alpha | + (1-a) |\beta\rangle \langle \beta|$$
We need to construct an orthonormal basis to, in which this system will act as a 2-level system. One of the variants is $$|+\rangle = |\alpha\rangle; \quad |-\rangle = \frac{|\beta\rangle - k|\alpha\rangle }{\sqrt{1-k^2}}, $$  where $k=\langle \alpha |\beta \rangle$.
The elements of the new density matrix $\rho^\pm$ are $$\rho_{11}=\langle+|\rho|+\rangle; \quad \rho_{12}=\langle+|\rho|-\rangle; \quad \rho_{21}=\langle-|\rho|+\rangle; \quad \rho_{22}=\langle-|\rho|-\rangle;$$
And thus it is:
$$\rho^\pm=\begin{pmatrix}a+(1-a)|k|^2 & \frac{k(1-a)(1-|k|^2)}{\sqrt{1-k^2}} \\ \frac{k^*(1-a)(1-|k|^2)}{\sqrt{1-k^{*2}}} & (1-a)(1-|k|^2) \end{pmatrix}.$$
It is possible now to calculate the entropy. When $|\alpha \rangle $ and $|\beta \rangle$ have the same phase, the dependence of the entropy on the parameter $a$ and separation between states $d$, $\langle \alpha |\beta \rangle = \exp{(-d^2)}$ looks like this:

It seems reasonable as it is zero at zero separation, as the state is pure than and also goes to $0$ when $a = 1$ or $0$. 
Edit: Thanks to Jess Riedel for the instructions.
