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.



Existence of a projection operator onto a classical set of density matrices

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> Can we construct a linear projection operator P onto C?
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No. The range of any linear operator will be a linear subspace.



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> If not, is there a nonlinear projection operator and if so how would one construct it?
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Yes, if $K$ is a closed convex subset of a Hilbert space $H$ there is a standard "projection" map $P: H \to K$ defined by letting $Pv$ be the closest element of $K$ to $v$. I guess the basic properties of this map are that $Pv = v$ for any $v \in K$ and $\|Pv - Pw\| \leq \|v - w\|$, i.e., $P$ is a contraction.


