Applications of Positive Operator Valued Measures (POVMs) I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator as an integral over a spectral measure. Because it is not that much extra work, he treats operator valued integrals in generality, allowing for integration over a POVM. As it is however, I can't find any examples or motivations for integrating over POVMs.
In quantum mechanics I have come across using POVMs to represent the most general form of measurement, but in that case a POVM is defined as a series of positive operators which sum to the identity. I suppose if you allow for a continuous range of results then this sum would become an integral, but is there anything else to it?
 A: If $A$ is a bounded linear operator on a complex Hilbert space $H$ such that $\|A\| \le 1$, then there exists a positive operator measure $P$ on the unit circle $T$ such that
$$
           A^n = \int_{T} z^n dP(z),\;\;\; n=1,2,3,\cdots.
$$
Note that $P(T)=I$. This becomes an abstract moment problem.
If $A : \mathcal{D}(A)\subseteq H\rightarrow H$ is the generator of a contractive $C_0$ semigroup $T(t)$ iff there is a positive operator measure on $[0,\infty)$ such that
$$
              T(t)f = \int_{0}^{\infty}e^{-ts}dP(s)f.
$$
Note that $P[0,\infty)=I$.
A positive Borel operator measure on $\mathbb{C}$ dilates to a spectral measure. More precisely, if $P$ is a positive operator measure on $\mathbb{C}$ with $P(\mathbb{C})=I$, then there exists an auxiliary Hilbert space $K$ and an isometry $V:H\rightarrow K$ and a spectral measure $E$ with values in $\mathcal{L}(K)$ such that
$$
                     P(S) = V^*E(S)V.
$$
If $P(\mathbb{C}) \ne I$, you can renormalize to use the above result.
If $F$ is a positive operator harmonic function on the open unit disk $D$, then there is a positive operator measure $P$ on the unit circle $T$ such that
$$
              F(re^{i\theta})=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-r^2}{1-2r\cos(\theta-\theta')+r^2}dP(\theta'), \\
   \mbox{ or } F(z) = \int_{T}\Re\left(\frac{z+w}{z-w}\right)dP(w)
$$
Using the dilation theorem, you can write $F$ as an operator expression involving the unitary operator $U=\int zdE(z)$, where $V^*EV=P$ and $E$ is a spectral measure. You find that $F$ is a positive operator function on the open unit disk $D$ iff there exists a unitary operator $U$ and $V : H \rightarrow K$ such that
$$
         F(z)  = \Re V^* \frac{zI+U}{zI-U}V
$$
That includes all scalar positive harmonic functions on the unit disk, $D$. Operator theory offers an interesting way to study harmonic functions, and these are equivalent to integral representations with respect to positive operator measures.
