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I am having a very hard time understanding exactly what a POVM is.

Could anyone provide a simple explanation?

Thank you

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- Thread starter Trixie Mattel
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- #1

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I am having a very hard time understanding exactly what a POVM is.

Could anyone provide a simple explanation?

Thank you

- #2

Nugatory

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The complete set of projectors corresponding to the eigenstates of an observable constitutes a projection valued measure (PVM).

The most general kind of measurement you can do, is to measure something correlated to the thing you want to observe. If observables [itex]X_{A}[/itex] and [itex]X_{B}[/itex] of particles [itex]A[/itex] and [itex]B[/itex] are highly correlated, you can "measure" [itex]X_{A}[/itex] by doing a standard measurement of [itex]X_{B}[/itex].

Unlike standard measurement, these indirect measurements of [itex]A[/itex] do not project the state of [itex]A[/itex] onto an eigenstate of [itex]X_{A}[/itex]. They cannot be described with a family of projections, but can be described with a corresponding family of positive operators. This family corresponds to a positive-operator-valued measure (POVM).

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This is quite different from a "standard" or ideal measurement, in which we know with certainty that if the system is in an eigenstate of the observable, we will definitely find the result given by the eigenvalue. That can be represented by a set of operators that have only 0 or 1 as eigenvalues- namely, projectors. Then the only sources of randomness in the result are the Born Rule probabilities when the state is in a superposition of eigenstates, and/or our lack of knowledge about the state, represented by

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- #6

bhobba

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In the standard quantum formalism a von-Neuman observation is a resolution of the identity where the elements are disjoint - each possible outcome can be mapped to an element of the resolution of the identity. It is also easy to form a Hermitian operator from such a resolution, and conversely given a Hermition operator, find its resolution of the identity. Such operators are of course the QM observables, but the real key is the resolution of the identity - the eigenvalues are simply real numbers we arbitrarily associate with each outcome. All this is carefully explained in Von-Neumann's classic - Mathematical Foundations of QM. A POVM simply removed the disjoint requirement. In many ways its more natural since you look at that requirement and say - why is it there? It's simply so you can form Hermitian operators. Big deal from a fundamental viewpoint. In fact it simplifies considerably a very important theorem - Gleason theorem (see post 137):

https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Physically it comes about given a system and one wants to do a von-Neumann observation on it. You introduce some kind of probe to do that observation. It turns out the observation you do on the probe is no longer a von-Neumann observation - its a POVM:

http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

Thanks

Bill

https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Physically it comes about given a system and one wants to do a von-Neumann observation on it. You introduce some kind of probe to do that observation. It turns out the observation you do on the probe is no longer a von-Neumann observation - its a POVM:

http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

Thanks

Bill

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