Simple example of a POVM measurement that is not projective

[Demystifier]

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

 

[Jazzdude]

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.

 

[Demystifier]

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

 

[bhobba]

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

 

[Jazzdude]

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

 

[A. Neumaier]

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.

 

[Demystifier]

Thank you all. The example by Neumaier satisfies my criteria the best.