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As part of my studies and research, I have learned to work with three different measurement formalism which I define to avoid any ambiguity with the nomenclature:

  1. General measurements, which are described by a collection of self-adjoint operators $\{M_m\}_m$ satisfying $\sum_m M_m^\dagger M_m= I$.
  2. POVMs, which are described by a collection of positive semi-definite operators $\{E_m\}_m$ satisfying $\sum_m E_m = I$.
  3. Projective measurements, which are described by a collection of positive semi-definite operators $\{P_m\}_m$ satisfying $\sum_m P_m= I$ and $P_m P_{m'}= \delta_{m,m'}P_m$.

As I am from a theoretical computer science background, I have but a limited understanding of the physical implementation of quantum measurements. I know that all three formalism are equivalent by using ancilla states and unitary evolution, however this could be hard to realise on a physical implementation of a quantum system. My question therefore is:

Is there a measurement formalism that is (in general) easier to realise across all known physical implementation of quantum states?

My intuition leads me to believe that projective measurements are the simplest to implement as they are they are a special case of the other two (when no ancilla is available), but this intuition might be flawed. As mentionned in Nielsen and Chuang, projective measurements are non-destructive, but there are measurement devices that destroy the state in the process of measuring it.

A maybe easier question, but one to which the answer would still satisfy me would be: what kind of measurement operator (i.e. formalism 1, 2 or 3) are being realised by devices that measure qubit systems.

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