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I really can not understand it. For instance, if a machine measures the quantum system. But no one checks this machine. Then the wave function of the quantum system is collapse or not? Besides, is measurements associated with photons?

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QM measurements are always measurements of specific observables. And a measurement always irrevocably alters the system that is being measured.

You are saying that the measuring device measures the QM system (and irrevocably alters its state), but no one (you mean person) checks the machine.

If a photon truly goes through both slits (at the same time), then why can't we detect it at both slits (at the same time)?

You are asking whether in this case the wave function collapses or not. There are many interpretations of QM, and wave function collapse (physical or not physical) is just one of the interpretations (Coppenhagen).

But what you really are asking is whether the QM system gets altered or not by the measurement itself, even if no one (you mean person) checks the measurement device. The answer is yes, because the measurement itself alters the state of the QM system.

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