Quantum Measurement: Sharp vs Unsharp - How to Determine?

[naima]

How can we see if a quantum measurement is sharp or unsharp ? can we measure unsharpness?
Thanks

 

[atyy]

I think there was some discussion over this by various groups. In the case of simultaneous measurements on non-commuting observables, the measurement of each is unsharp. How should one quantify the unsharpness?

http://arxiv.org/abs/1208.0034
http://arxiv.org/abs/1306.1565

 

[naima]

Could you look at this http://www.johnboccio.com/research/quantum/notes/Operational_QM.pdf
http://www.johnboccio.com/research/quantum/notes/Operational_QM.pdf
Busch gives a formal definition of unsharpness on page 36. I do not undestand what is the physics behind that. Could you comment.
The author writes Eu (H) = Ep (H)\P(H)
So unsharp measurements have outputs that are non projector effects.
The question is now: How can i recognize a non projector?

 

[naima]

I am beginning to see how it works.
In operational quantum mechanics, the outcomes of an apparatus are no more associated to vectors in H but to operators on H.
take an optical device with one input channel receiving linear polarized photons and with 3 output channels associated to
\begin{pmatrix}
cos^2(\alpha) & 0 \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
sin^2(\alpha) & 0 \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
They sum to Id
There is a difference between the three outputs:
If you prepare down photons and send them thru the device the third detector detector 1 and 2 will not click and the 3th will click. we have here a sharp detection.
We cannot prepare a state so that the only first detector will click detectors 1 and 2 are associated to unsharp measurement (unless alpha 0)
if $\alpha = \pi /2$ we have a maximal unsharpness for detectors 1 and 2.
they will have a 1/2 probability to click for an up photon