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I am thinking in following way of thinking about measurements in quantum mechanics. Please correct any false statements I may be making below.

  1. We start with POVMs. Let our POVM be a set of positive operators $\{E_i\}$ that sum to the identity. We have a set of possible outcomes $i$ and each outcome occurs with probability $p_i$. The post-measurement state depends on the implementation of the POVM so we cannot say anything about it.

  2. We could rewrite the POVM as $\{E_i = M^\dagger_i M_i\}$ - a specific implementation of the POVM. Now, we can also say that if we observe outcome $i$, we have a post measurement state $\rho = \frac{M_i\rho M_i^\dagger}{\text{Tr}(M_i\rho M_i^\dagger)}$.

  3. The last case is where the $M_i$ are projectors. Now, we can repeat measurements and the state (after the first collapse) stays in the eigenstate that it collapsed to.

Note that this formalism says nothing about eigenvalues. All you have are states, outcomes that occur with a certain probability and post-measurement states. We could label these outcomes with something and call the labels the result of the measurement.

On the other hand, we often hear this statement that all quantum observables correspond to Hermitian operators and their real eigenvalues represent the "measurement" we've made. How does one connect this to the formalism I wrote above? I can see that every Hermitian observable can be written as a projective POVM with some real number attached to each outcome. An answer here explains that in principle, one can label the POVM outcomes with anything (even apples, bananas and oranges as suggested).

Questions

  1. Do we choose Hermitian operators because we'd like all measurements we make in practice to have a real number in the outcome label and would like the state to collapse to one among an orthogonal set of eigenstates?
  2. If I chose not to use real numbers, do I not lose the physical meaning of quantities like energy? A three level atom has energy levels $E_1, E_2$ and $E_3$ and no matter what scale I use, the difference $E_2 - E_1$ being greater than or less than or equal to the difference $E_3 - E_2$ has clear physical consequences, no? It therefore seems like real eigenvalues are necessary if one wants to measure things like energy, position, etc.
  3. What about POVMs of type 1. and 2. above? Why are physical observables like position or momentum not of type 1 or 2?
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  • $\begingroup$ Since type 3 (you are refering to your first list labeling types?) Is a subset of type 2 and type 2 is a subset of type 1; physical observables are of course type 3, 2 and 1. $\endgroup$ – lalala Feb 5 at 20:10
  • $\begingroup$ I know of a sneaky trick that allows you to measure complex-valued facts about quantum systems: you can measure the real and imaginary parts separately. They are both real numbers, and so must have Hermitian operators (after all it wouldn't make sense to have a complex eigenvalue to represent them). If you believe that argument, you will only ever need Hermitian observables, even if for some reason the number you want to measure is "complex." $\endgroup$ – Display Name Feb 5 at 20:47
  • $\begingroup$ @lalala however, there exist POVMs in type 2 which are not type 3. Why don't we ever associate observables with these? $\endgroup$ – user1936752 Feb 6 at 9:37
  • $\begingroup$ @DisplayName yes, but I suppose I'm having trouble with the converse. If I want to measure real valued facts, wouldn't any operator that has non real eigenvalues inherently fail at describing the system correctly? If so, how does this square with the answer linked in my answer that Hermitian observables hold no special place in the theory of measurement? $\endgroup$ – user1936752 Feb 6 at 9:39

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