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Quantum Measurement can be divided into General Measurements, Projective measurements and general POVMs. And there are also some special kinds of quantum measurements that have their own name, such as :

  • adiabatic measurement (protective measurement)
  • weak measurement
  • quantum nondemolition measurement and so on.

Is there a list of those special kinds of measurements?

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  • $\begingroup$ note that "general measurement" and "general POVM" are pretty much the same thing (though in the "general measurement" formalism one often keeps track of/defines post-measurement states). More generally, any measurement can be thought of as a special case of a POVM/general measurement. $\endgroup$
    – glS
    Apr 27 at 10:05
  • $\begingroup$ In the Book of Nielsen and Chuang, they describe them in two sections. But I think it's the reason for writing an easy understanding book. But in box 2.5, they stated "POVMs are best viewed as a special case of the general measurement formalism, providing the simplest means by which one can study general measurement statistics, without the necessity for knowing the post-measurement state." $\endgroup$
    – narip
    Apr 27 at 10:24
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    $\begingroup$ see also the relevant discussion in quantumcomputing.stackexchange.com/q/12275/55 $\endgroup$
    – glS
    Apr 27 at 10:27
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    $\begingroup$ A POVM alone is only able to describe the measurement statistics. To describe post-measurement states, we have to go beyond POVMs and define the actual quantum channel corresponding to the measurement. This is sometimes called a "quantum instrument". Note that many quantum instruments correspond to the same POVM. This is due to the non-uniqueness of a "square root" $M$ for a given effect $E=M^\dagger M$. $\endgroup$ Apr 27 at 11:55

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