Understanding POVM: A Simple Explanation

[Trixie Mattel]

Hello,

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

Could anyone provide a simple explanation?

Thank you

 

[Nugatory]

You will get better answers if you can point to the the definitions that you've already seen, and tell us where you're having trouble understanding.

 

[jfizzix]

In the standard case, the act of measurement projects the quantum state onto an eigenstate of the observable measured.
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).

 

[maline]

The way I understand it (I'm not too sure though) is that a POVM is a set of operators, each of which represents a possible result in your experiment. Unlike the operators for "observables", in which the eigenvalues are the physical values that may be found in the measurement (with units!), these operators have eigenvalues that are numbers between zero & one. The physical meaning of the eigenvalue is that if the measured system is in the associated eigenstate, then the result represented by this particular operator will occur with a probability given by the eigenvalue. More generally, if the system is in state ρ, then the result represented by operator E will occur with probability Tr(ρE).
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 ρ being mixed. The POVM formalism allows us to describe more realistic measurement scenarios, in which there usually are many other probabilistic factors at play as well.

 

[maline]

I should also mention the condition that the set of POVM operators must sum to the identity. This simply represents the constraint that for any state ρ, the probabilities for all possible results must sum to one. If your set sums to something less then the identity, meaning that the difference is itself an operator with eigenvalues between 0 and 1, then you simply append that operator to the set. It represents the null result- none of your detectors went off, the particle got lost, whatever.

 

[bhobba]

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