Operator as the outcome of a measurement?

[Heidi]

TL;DR Summary

POVM?

for any set of POVM outcomes it is possible to construct a setup with say one incoming photon and possible outcomes that will click differently. so this is not only mathématics.
but what is physically an operator valued measurement?

 

[Heidi]

you can see how to implement such a device for any povm
https://www.researchgate.net/publication/2193434_General_implementation_of_all_possible_positive-operator-value_measurements_of_single_photon_polarization_states

Maybe the random states obtained after a measuremt are naturally described in the C* algebra language.

 

[Morbert]

Are you asking if there is some further generalisation of measurement? PVM -> POVM -> ?

 

[Heidi]

No,
when i have a measurement device i am looking for a (maybe random) value associated to an obsevable. In a Stern Gerlach i get a click in a path associated to a spin up or down.
Look at the implementation in the link. we have a brick with two output that you can associate like a lego. and the "click may appear
in the different spots. i know that each of them corresponds to a POVM outcome but in what is it a physical result?

 

[Heidi]

Even if i do not see what is measured, it gives me a recipe to prepare any state.

 

[Morbert]

Ok I think I understand what you are asking. Given a POVM $\sum_iF_i = \sum_iM^\dagger_iM_i = I$, we can interpret the associated outcomes $\{O_i\}$ a few different ways:

i) If there is some observable $A = \sum_i a_i E_i$ such that $$\frac{M_i\rho M^\dagger_i}{\mathrm{tr}\left(M_i\rho M^\dagger_i\right)} \approx \frac{E_i\rho E_i}{\mathrm{tr}\left(E_i\rho\right)}$$then we can still say our device measures $A$ even if this is approximate. Whether or not this approximation is sufficient will depend on your experimental resolution. It's why we can say e.g. that an SG device still measures spin even though strictly it is a POVM. (Also note the dependency on the preparation here)

ii) More generally, for a given preparation $\rho$ and outcome $O_k$ we can write down at least one observable $A_k = \sum_i a_{ik} E_{ik}$ such that $\rho$ and $O_k$ assign a probability for each possible value of the observable $a_{ik}$

 

[Heidi]

In the link we get F1 =
$$\begin{pmatrix}
cos^2 \alpha & 0 \\
0 & cos^2 \beta
\end{pmatrix}$$

and F2 =
$$\begin{pmatrix}
sin^2 \alpha & 0 \\
0 & sin^2 \beta
\end{pmatrix}$$
What is the measured observable?

 

[Morbert]

In the link we get F1 =
$$\begin{pmatrix}
cos^2 \alpha & 0 \\
0 & cos^2 \beta
\end{pmatrix}$$

and F2 =
$$\begin{pmatrix}
sin^2 \alpha & 0 \\
0 & sin^2 \beta
\end{pmatrix}$$
What is the measured observable?

PVMs are associated with observables, POVMs are not so there is no singular measured observable. Instead, the outcomes of POVMs are associated with observables (see my last post for details). So e.g. The different $F$s in that paper (equations 18-21) can be associated with different angles of polarisation, and so each $F$ is associated with a different observable.