A POVMs for Infinite Dimensional Hilbert Spaces

jbergman
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Most discussion of POVM focus on examples on finite-dimensional Hilbert Spaces. How do we construct a POVM for multiple observables with continuous spectrum?
After reading up on some of the discussion in the Quantum Interpretations forums, I became interested in learning more about POVMs.

However, most of the examples are from the finite dimensional setting. If I wanted to model a POVM that approximately measures position and momentum simultaneously, how would I construct such a POVM?
 
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See Part IV https://journals.aps.org/pra/pdf/10.1103/PhysRevA.87.062112.
 
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You could search for "Arthurs-Kelly" measurements.

Focusing in Arthurs-Kelly-type Joint Measurements with Correlated Probes
Thomas J Bullock, Paul Busch
https://arxiv.org/abs/1405.5840

Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol
Maicol A. Ochoa, Wolfgang Belzig & Abraham Nitzan
https://www.nature.com/articles/s41598-018-33562-0
 
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Since measurements always produce discrete results, finite dimensions are quite appropriate for POVMs in practice; see the paper accompanying my Insight article Quantum Physics via Quantum Tomography.

Many examples of POVMs in infinite dimensional Hilbert spaces arise from them by taking a continuum limit.
 
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A. Neumaier said:
Since measurements always produce discrete results, finite dimensions are quite appropriate for POVMs in practice; see the paper accompanying my Insight article Quantum Physics via Quantum Tomography.
A finite valued POVM isn't the same as a POVM that acts on state vectors in finite dimensional Hilbert space is it?

For instance, if I want to measure a discrete position value.
 
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jbergman said:
A finite valued POVM isn't the same as a POVM that acts on state vectors in finite dimensional Hilbert space is it?
Yes, there is a difference. But for finite valued POVMs, the (finite or infinite) dimension of the Hilbert spaces does not really figure in the formalism.
jbergman said:
For instance, if I want to measure a discrete position value.
In this case you first need to define what you mean by discrete position. Maybe you find the setting in Sections 3.3-4 of my quantum tomography paper
  • A. Neumaier, Quantum mechanics via quantum tomography, Manuscript (2022). arXiv:2110.05294v3
convincing.
 
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