# Interpretation of a density matrix as an observable

In quantum mechanics, any density matrix (or density operator) is Hermitian. Observables are also represented by Hermitian operators. So it follows that a density matrix can also be interpreted as an observable.

What is the physical meaning of this observable?

In the simple case of a pure state, the density matrix is the projection $\rho_\psi=\lvert\psi\rangle\langle\psi\rvert$, for some $\psi\in \mathscr{H}$ (the Hilbert space). Measuring it on another general state $\rho$ (i.e. calculating $\mathrm{Tr}[\rho_\psi\rho]$) gives you the probability of $\psi$ being the "configuration" (or Hilbert space vector) corresponding to $\rho$.
Not every Hermitian operator is an observable. The identity on a space is certainly Hermitian, but it is no observable in any physical sense - its eigenvalue is $1$, and that's it. The eigenvalue of scalar operators gives you precisely zero information about the state considered, and usually, we would like an observable to at least give us some information about the state we measure.
• @yuggib: Right. As so often, I'm partial to the nLab approach of generating the $C^*$-algebra of observables from smooth functions on the classical phase space. In this approach, it would be unequivocally clear that operators that do not come from functions on the phase space (and such an operator is the density matrix, I think), are not to be called observables. However, as you have shown, it is possible to take another, broader, stand on observables. – ACuriousMind Jul 20 '14 at 14:29
• @celtschk: Alright, if you see it that way, the identity is observable - but is it really meaningful to call something an observable if its observed value does not depend on the state measured at all? And you are right, the constant function with value $1$ seems to me to produce indeed the identity operator. I've spoken too fast. It all boils down to what you consider to be "physically observable", see also this old question, where they could not reach a real consensus. – ACuriousMind Jul 20 '14 at 15:09