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Eigenvalue Postulate and Experiment Outcomes in QM

In Nielsen and Chuang's text on Quantum Information and Computation, the measurement postulate is stated by using a collection of measurement operators and the outcomes are the indices of the ...
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Diff(M) and requirements on GR observables

This question is kind of inspired in this one: Diff(M) as a gauge group and local observables in theories with gravity The conundrum i'm trying to understand is how is derived the (quite) ...