POVMs for Infinite Dimensional Hilbert Spaces

[jbergman]

TL;DR

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?

 

[Haborix]

See Part IV https://journals.aps.org/pra/pdf/10.1103/PhysRevA.87.062112.

 

[atyy]

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

 

[vanhees71]

See Part IV https://journals.aps.org/pra/pdf/10.1103/PhysRevA.87.062112.

Here's the arXiv version:

https://arxiv.org/abs/1211.4169v4

 

[A. Neumaier]

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.

 

[jbergman]

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.

 

[A. Neumaier]

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.

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.