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Reading the "projective measurements" section in Nielsen and Chuang, I'm wondering how eigenvalues are actually relevant.

The setup has a Hermitian operator $M = \sum_m m P_m$ where $P_m$ is the projector on the eigenspace of $M$ with eigenvalue $m$. Then the probability of "getting result $m$" is $p(m) = \langle \psi | P_m | \psi \rangle $.

So it seems that the eigenvalues are really just used as a form of labelling, and their actual value is irrelevant. Is this so? And is this the reason we are considering Hermitian operators instead of normal operators?

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  • $\begingroup$ If I have operator in the form $\sum m P_m$ then in a sense yes the eigenvalues are just labels. If, however I have my operator in some other form, say for example $H = \frac{p^2}{2m} + \frac{1}{2}m\omega x^2$ then knowing that the possible results when I measure $H$ are the eigenvalues of $\frac{p^2}{2m} + \frac{1}{2}m\omega x^2$ is extremely useful information. $\endgroup$ – By Symmetry Oct 28 '16 at 22:03
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The eigenvalues are really just a form of labeling and their actual value is irrelevant. This has nothing to do with quantum mechanics. The temperature of your swimming pool is also just a form of labeling --- you can measure it in degrees Fahrenheit, or degrees Kelvin, or the square of degrees Fahrenheit, or the square root of the exponential of the degrees Kelvin, and you'll be conveying the same information no matter which you choose to report. Ditto for everything else you'll ever measure, quantum mechanically, classically, or anywhere else.

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The actual measurements are perfomed with measurement devices. That is that those outcomes are correlated with certain values of some macroscopic degrees of freedom, most stereotypically with position of a pointer. Because those macroscopic degrees of freedom can be described by real numbers, you end up with hermitian or reducible to hermitian observables.

For Hamiltonian there exist extra restriction. It's generator of the evolution in time that should be unitary and therefore the hamiltonian itself should be hermitian.

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The eigenvalues become relevant once you employ some sort of the quantum correspondence principle, like Weyl quantization. In such cases, the requirement is that the results of measuring the same observable (like energy) should be comparable in quantum and classical physics, and a part of that is agreeing in magnitude and dimension. It does not affect probabilities as values, but affects what events the probabilities are assigned to.

For an observable like Pauli Z in quantum information this indeed does not affect much, and the interpretation or realization of the measurement does not change if we take any observable of the form $a + b\sigma_z$ instead. But if we speak of a spin-Z measurement, it must be of the form $$S_z = \frac ħ2σ_z$$ and give values of $+ħ/2$ ($|\!↑〉$) or $-ħ/2$ ($|\!↓〉$) with the dimension of action, so that it can be directly compared with other angular momenta.

If correspondence to real world measurements is not important to you, there have been some attempts at relaxing the Hermiticity condition, whose main use is to guarantee a real-valued spectrum. This relatively popular article, for example, suggests using complex units instead for observables which are naturally 2π-periodic (thus describing measurements using unitary operators instead of Hermitian), and goes as far as claiming that not even normality is necessary.

If you only are interested in the probabilities, and are happy to assign them to abstract events, you can ditch the operator observables entirely and introduce some more general concept like positive operator-valued measurements instead, which do not describe the measurement using any single operator. I think it's also done in your book, just somewhat later.

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