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.

  • $\begingroup$ FYI, you should use \langle and \rangle rather than < and >, as they are interpreted differently by the formatting engine. $\endgroup$ – MTS Apr 26 '13 at 18:55
  • $\begingroup$ 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. $\endgroup$ – 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.

| cite | improve this answer | |
  • $\begingroup$ 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? $\endgroup$ – Sebastian Meznaric Apr 27 '13 at 11:46
  • 1
    $\begingroup$ @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. $\endgroup$ – Nik Weaver Apr 27 '13 at 15:56
  • $\begingroup$ Yeah, that settles that. Thanks again for your help. $\endgroup$ – 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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.