A Does the existence of a POVM require an ancilla?

  • A
  • Thread starter Thread starter jbergman
  • Start date Start date
  • Tags Tags
    Existence
jbergman
Messages
481
Reaction score
221
TL;DR Summary
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?
 
  • Like
Likes Son Goku and Demystifier
Physics news on Phys.org
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.
 
Last edited:
  • Like
Likes atyy, jbergman and vanhees71
jbergman said:
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.
 
  • Like
Likes jbergman, atyy and vanhees71
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top