In order to verify experimentally the quantum mechanical predictions when we measure observable $\hat O$ on a given system A, it is usual to prepare an ensemble of identically prepared systems ($N$ number of A systems, for large $N$) and subject each member of the system to the same measurement $\hat O$. Does the Hilbert space corresponding to this ensemble always equal the tensor product of the Hilbert spaces of each member of the ensemble? Moreover, it is also known that a single experiment on a single system is experimentally meaningless in quantum mechanics (except when the single system was in one of the eigenstates and the measurement of the corresponding observable is made). Does the affirmative answer (if applicable) to the first question and the above-mentioned fact, imply that it is always necessary to use a Hilbert space which is the tensor product of the Hilbert spaces of the set of identically prepared systems, in order to verify experimentally the predictions made by quantum theory?


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