I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as $$ \rho = \int d^2\alpha \ P(\alpha) |\alpha\rangle \langle\alpha|.$$

The states $|\alpha\rangle$ form an overcomplete set and are not all orthogonal to one another. If $P(\alpha)$ is positive then we have a classical mixture of coherent states, and so we call such a state classical. Given two classical states $\rho$ and $\sigma$, it then follows that a convex combination $p \rho + (1-p) \sigma$ is also a classical state. So the set of classical states, let's call it $C$, forms a convex cone.

Now to the question. Can we construct a *linear* projection operator $P$ onto $C$? If not, is there a nonlinear projection operator and if so how would one construct it? I am most interested in constructing this operator, but if you can give me a list of interesting properties that would also be welcome.