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I know the definition of projective measurement, generalized measurement, POVM.

I understand the usage of generalized measurement for the reason that it can model experiments "easier" (for example measurement of a photon that will be destructive so that measuring again the state just after the first measurement will give me another answer).

However, I am still kinda confused by why we have introduced the notion of P.O.V.M. For me we have everything we want from generalized & projective measurement.

Would you agree with me if I say that POVM is just an **axiomatic** way to define statistics of measurement. There is nothing much to understand/overthink.

In the sense, we ask the minimal mathematical properties that our measurement operator must fullfil with respect to statistical behavior which is:

- they are semidefinite positive (to have positive probabilities)
- they sum up to identity (to have probability summing up to $1$)

and we relate our measurement operator with the physics:

$$p(m)=\mathrm{Tr}(E_m \rho)$$ where $m$ is the outcome, $E_m$ the associated POVM.

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For a start, see Martin's answer here.

– Sanchayan Dutta – 2020-01-06T16:57:26.837