How to teach beginners in quantum theory the POVM concept

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Discussion Overview

The discussion focuses on how to effectively teach beginners the concept of Positive Operator-Valued Measures (POVMs) in quantum theory. It explores pedagogical approaches, the introduction of quantum states, and the derivation of measurement principles, particularly in the context of qubits and their representation through classical light polarization.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that introducing POVMs for physicists requires using standard formulations of observables and states, arguing that it is simpler than introducing Born's rule in full generality.
  • Another participant proposes that the measurement postulate can be derived from a principle called the Detector Response Principle (DRP), which relates detector responses to the state of the source.
  • There is a discussion on how to derive the properties of matrices associated with POVMs from experimental observations, emphasizing linearity and the operational determination of coefficients through quantum detection tomography.
  • Some participants express curiosity about how the concept of a state as a trace class operator is motivated, particularly in relation to the qubit model introduced in the Insight article.
  • Questions arise regarding the generalization of the qubit model to arbitrary Hilbert spaces and the justification for applying this formalism to all physical systems beyond photons.
  • One participant highlights that the Insight article effectively leads to the understanding of complex positive semidefinite Hermitian operators as representations of quantum states.

Areas of Agreement / Disagreement

Participants express varying levels of agreement on the effectiveness of the Insight article in introducing quantum concepts, but there is no consensus on how to generalize these concepts to broader physical systems or the motivation behind the trace class operator representation.

Contextual Notes

There are unresolved questions regarding the assumptions made in the transition from qubit models to general quantum systems, particularly concerning the tensor product structure and the applicability of the formalism to different physical contexts.

  • #61
Seems to be a great book! I've even (legal) access to it, though it's published in 1995. Is this the book you have in mind? At least it's about the right topic and also discusses the SGE in some detail.

https://rd.springer.com/book/10.1007/978-3-540-49239-9
 
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  • #62
vanhees71 said:
Seems to be a great book! I've even (legal) access to it, though it's published in 1995. Is this the book you have in mind? At least it's about the right topic and also discusses the SGE in some detail.

https://rd.springer.com/book/10.1007/978-3-540-49239-9
Yes, this is the book I had quoted from. There is also another, quite recent book with two authors the same:
  • P. Busch, P. Lahti, J. Pellonpää and K. Ylinen, Quantum Measurement, Springer, Berlin 2016.
with a number of chapters on realistic POVM measurements, but the other book is much more elementary.
 
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  • #63
vanhees71 said:
At least it's about the right topic and also discusses the SGE in some detail.
Here is another book of interest:
Chapter 7 discusses a number of realistic examples. In the introduction to the chapter (p.258 of the online version) he writes:

Willem de Muynck said:
The examples discussed in sections 7.2 through 7.5 show that a generalization of the formalism is necessary for describing even the most common methods of quantum mechanical measurement, like the detection of photons using a detector that is not 100% efficient. This also holds true for such experiments as the double slit experiment, being a paradigm of standard quantum mechanics. It will not be surprising, then, that an analysis of this experiment based on the standard formalism can hardly be a reliable one, and that conclusions based on such an analysis should be considered with some reservation.
With ''generalization of the formalism'' he means the POVM formalism generalizing the traditional textbook formalism which only features projective measurements.
 
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