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Jun 3, 2021 at 16:52 comment added QuestionEverything You are partially right regarding the post-measurement. This is sort of a philosophical issue you know. What we observe from the measurement of a quantum state is indeed dictated by our own measurement apparatus. But briefly, yes, it refers to the post-measurement state. Your second question: yes, the labels are arbitrary, only the actual operators are important.
Jun 2, 2021 at 17:46 comment added theQman Your question: Is the system in the |0⟩ state or in the |1⟩ state? This refers to the system post-measurement, right? As for my confusion of the measurement labels, I thought the index labels were arbitrary. Can't we call the operators $M_0, M_1$, say, $M_{cat}, M_{dog}$, in which case, after applying our $\{M_{cat}, M_{dog}\}$ measurement, we get a "cat" outcome leaving the system in state |0⟩, or a "dog" outcome leaving the system in state |1⟩, each with equal probability?
Jun 2, 2021 at 16:52 history answered QuestionEverything CC BY-SA 4.0