I want to understand positive operator valued measures in QM

[Fredrik]

Main Question or Discussion Point

I want to understand positive operator valued measures in QM, in particular why they are considered "observables". Anyone know a good place to start reading about this?

I know some functional analysis and some measure theory, but I haven't made it all the way to the spectral theorem.

 

[A. Neumaier]

I want to understand positive operator valued measures in QM, in particular why they are considered "observables". Anyone know a good place to start reading about this?

I know some functional analysis and some measure theory, but I haven't made it all the way to the spectral theorem.

Chapter 3 of
http://www.theory.caltech.edu/people/preskill/ph229/#lecture

 

[Fredrik]

Thank you. That looks like a good enough place to start.

 

[A. Neumaier]

Thank you. That looks like a good enough place to start.

The previous was just the first convenient online reference I found.
After a more systematic search, I recommend the following:
http://arxiv.org/pdf/quant-ph/0611067
http://arxiv.org/pdf/quant-ph/0608087
https://wiki.physik.uni-muenchen.de/TMP/images/8/87/POVMs.pdf

 

[kith]

What does "positive operator valued" or "projector valued" measure mean? Is this really an accurate naming?

I'm asking, because the possible outcomes in such measurements are still eigenvalues of a suitable operator, aren't they?

 

[A. Neumaier]

What does "positive operator valued" or "projector valued" measure mean? Is this really an accurate naming?

I'm asking, because the possible outcomes in such measurements are still eigenvalues of a suitable operator, aren't they?

No. outcomes can be anything; the POVM only specifies their probabilities:

the probability the outcome associated with measurement of operator F_i occurs is
P_i = tr rho F_i, where rho is the density matrix of the measured system.

(from http://en.wikipedia.org/wiki/POVM )

This is far superior to the heavily idealizing Born interpretation, which claims that a measurement produces the exact eigenvalue of an operator, which is ridiculous whenever the eigenvalues of the operator in question are irrational numbers. (In particular, discrete spectra of all but the simplest Hamiltonians are irrational, but nobody is able to measure energies to infinitely many digits!)

 

[dextercioby]

What does "positive operator valued" or "projector valued" measure mean? Is this really an accurate naming?

Yes, it's accurate. First note that <projector operator> is a well defined concept in a Hilbert space. They are operators which have certain particular properties for which they are useful in quantum physics (2 one of them being boundedness and the property to be positively defined). Also <measure> is a concept used in Hilbert spaces through spectral measures for self-adjoint/unitary operators. So one achieves the most general description of self-adjoint operators through projector valued measures (PVM )by the spectral theorem of von Neumann. For positively defined operators, such as the statistical operator ρ of von Neumann, we also have POVMs as most people know them again through the spectral theorem. Actually, POVMs generalize PVMs in the sense discussed in https://wiki.physik.uni-muenchen.de/...8/87/POVMs.pdf [Broken] page 4.

 

[A. Neumaier]

So one achieves the most general description of self-adjoint operators through projector valued measures (PVM )by the spectral theorem of von Neumann.

Not quite. An operator contains more information than recorded in the PVM; therefore the latter gives not a full description of the operator, and hence not the most general one:

If A is a self-adjoint operator with discrete spectrum and not itself a projector then the operators
A and A^2 have different spectrum but the same associated PVM.

 

[homeomorphic]

A good reference to start learning about the spectral theorem is Mathematical Physics by Robert Geroch, in the last few chapters of the book. He doesn't do the operator-valued measure approach, although he mentions it (I think it's still worth reading first, even if you are more interested in operator-valued measures). Very nice exposition, though.

 

[dextercioby]

OK, I stand corrected. Nonetheless, A and A² may not have the same domain...

 

[A. Neumaier]

OK, I stand corrected. Nonetheless, A and A² may not have the same domain...

They have the same domain iff A is bounded.

But A^2 and A^2+1 always have the same domain and the same PVM, but different spectrum.