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

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.

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FYI, you should use \langle and \rangle rather than < and >, as they are interpreted differently by the formatting engine. –  MTS Apr 26 '13 at 18:55
Yeah, thanks for fixing my formatting @MTS. @Uwe Franz, yes I would like to send the general states to classical states and leave the classical states invariant. The set of classical states in this sense is not a linear subspace but a convex subspace of a linear space. –  Sebastian Meznaric Apr 27 '13 at 11:44

Can we construct a linear projection operator P onto C?

No. The range of any linear operator will be a linear subspace.

If not, is there a nonlinear projection operator and if so how would one construct it?

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.

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This is exactly what I was looking for. This is what I was looking for, but do you also know if in case that for each $v \in H$ there is a unique closest element $x \in K$, can we say anything about uniqueness of a projection operator to $K$, or is this just one possible construction? –  Sebastian Meznaric Apr 27 '13 at 11:46
@Sebastian: You may ask whether the two conditions I listed uniquely characterize $P$. I don't think so. For example, take $H = {\bf R}$ and $K = [0,\infty)$. Then the map I described is $P(x) = x$ if $x \geq 0$ and $P(x) = 0$ if $x < 0$. But the map $P(x) = |x|$ would also have the two listed properties. –  Nik Weaver Apr 27 '13 at 15:56
Yeah, that settles that. Thanks again for your help. –  Sebastian Meznaric Apr 30 '13 at 13:21

H there is a standard "projection" map P: H \to K defined by letting Pv be the closest element of K to v.

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