Are POVMs fundamental or require Naimark's theorem and ancillas?

[jbergman]

TL;DR

It isn't clear to me whether POVMs are fundamental or are instead just derived from PVMs on a larger Hilbert space.

In DeMuynck's paper, POVMs: a small but important step beyond standard quantum mechanics, he describes a "generalized quantum mechanics" in which a generalized observable can be represented by POVM.

In contrast, most other references that I have seen discussing this talk about first constructing a PVM on a larger Hilbert space with ancilla state and then measuring a POVM.

I am confused about how fundamental POVMs are. Is it fair to postulate the existence of POVMs even without Naimark's theorem?

 

[Demystifier]

Excellent question!

Those two views of POVM measurements correspond to two attitudes on the measurement problem in QM. In one approach, dating back to Bohr, the measuring apparatus is treated as a classical object, or alternatively, the measurement is treated as a primitive notion that does not need to be derived from something more fundamental. In another approach, dating back to von Neumann, the measurement should be explained in terms of quantum states of the measuring apparatus. The POVM measurements are viewed as fundamental in the first approach, but derived from PVM's in a larger Hilbert space in the second approach.

 

[Son Goku]

Summary: It isn't clear to me whether POVMs are fundamental or are instead just derived from PVMs on a larger Hilbert space.

I am confused about how fundamental POVMs are. Is it fair to postulate the existence of POVMs even without Naimark's theorem?

Yes you can. Quantum Theory can be formulated as mixed states giving statistics for POVMs and evolving under CPTP maps.

Yes all of these can also be "purified" to pure states, PVMs and Unitaries respectively, but equally pure states can be seen as a special case of mixed states and the same goes for CPTPs and unitaries and PVMs and POVMs. This actually reflects an important high level symmetry of QM known as purification, also reflected in the fact that the category of quantum theory, QUANT, can be defined in two ways.

See D'Ariano's "Quantum Theory: An informational approach" for an intro to purification, or Greg Kuperberg's lecture notes here for an advanced treatment:
https://www.math.ucdavis.edu/~greg/intro-2005.pdf
In D'Ariano's axiomatic derivation of QM purification alone differentiates quantum probability from classical probability. His first five axioms are shared by quantum and classical probability and no other probability theory. Purification is the sixth axiom.

I think it's more important to realize this is a deep feature/symmetry of QM, rather than asking which of PVMs or POVMs are more fundamental.