POVM Measurement Example: Intuitive & Easy-to-Understand

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Discussion Overview

The discussion revolves around finding a simple, intuitive example of a Positive Operator-Valued Measure (POVM) measurement that is not a projective measurement. Participants explore various theoretical and practical aspects of POVM measurements, including their physical interpretations and examples from quantum mechanics.

Discussion Character

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

Main Points Raised

  • One participant requests a straightforward example of a POVM measurement that is not projective.
  • Another participant suggests using a tensor factor of a larger system and performing a projective measurement on that larger system, leading to a POVM measurement for the subsystem.
  • A participant expresses that while the mathematical explanation is clear, the physical interpretation remains unclear and asks for a practical example.
  • Another participant proposes measuring the total spin of a two-particle system with nonzero spin, describing it as a POVM measurement for each individual particle.
  • It is noted that measuring an entangled system in the Bell basis results in a POVM for each subsystem, although the practical execution of such measurements is acknowledged as uncertain by one participant.
  • A different example involving coherent states is mentioned, indicating that they represent simultaneous but unsharp measurements of position and momentum.
  • A participant expresses satisfaction with one of the examples provided, indicating it meets their criteria for clarity.

Areas of Agreement / Disagreement

Participants express varying degrees of clarity regarding the physical interpretation of POVM measurements. While some examples are proposed, there is no consensus on a single example that is universally accepted as intuitive or easy to understand.

Contextual Notes

Some participants highlight the challenge of connecting mathematical clarity with physical intuition, indicating a potential gap in understanding practical implementations of POVMs.

Demystifier
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I need a simple, intuitive, and easy-to-comprehend example of a POVM measurement, which is not a projective measurement. Any suggestions?
 
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Take a system which is a tensor factor of a larger system and perform a projective measurement on that larger system with a basis that does not separate in that tensor factor. The subsystem is then undergoing a POVM measurement.
 
Jazzdude, it is clear mathematically but not physically. How a measurement in such a mixed basis can be performed in practice? Any simple example?
 
Demystifier said:
Jazzdude, it is clear mathematically but not physically. How a measurement in such a mixed basis can be performed in practice? Any simple example?

Indeed - mathematically I love that explanation - but physically its meaning is unclear. Physically I think the following is better - see page 14-16:
http://arxiv.org/pdf/quant-ph/0205039v1.pdf

Its basically what results when you let a measuring system interact with the system being measured then observe the measuring system.

Also note the cool almost trivial proof of Gleasons Theorem with POVM's.

Thanks
Bill
 
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Demystifier said:
Jazzdude, it is clear mathematically but not physically. How a measurement in such a mixed basis can be performed in practice? Any simple example?

Take a system of two particle with nonzero spin and measure the total spin. Then you can describe the measurement as a POVM measurement for each individual particle.

In quantum information theory one often considers measuring an entangled system in the Bell basis, which also results in a POVM for each subsystem.

Does that make it clearer? If you're asking how you actually perform an experiment to measure those I must admit that I have almost no knowledge about that.

Cheers,
Jazz
 
Demystifier said:
I need a simple, intuitive, and easy-to-comprehend example of a POVM measurement, which is not a projective measurement. Any suggestions?

The family of coherent states is a POVM of the kind you want, describing a simultaneous but unsharp measurements of position and momentum.
 
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Thank you all. The example by Neumaier satisfies my criteria the best.
 

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