Let $|\psi\rangle = |\psi_1\rangle \otimes ... \otimes |\psi_n\rangle$, where $n \in \mathbb{N}, n\geq2$ and $|\psi_i\rangle \in \mathbb{C}^2$ is a quantum state, for all $i \in \{1,...,n\}$.

Let $i \in \{1,...,n \}$ be a random but fixed index. Suppose we want to measure only the qubit $|\psi_i\rangle$ of the system $|\psi\rangle$, and the (projective) measurement of that qubit is described by an observable $M$.

Can we say that the (projective) measurement of the system is described by the observable $I \otimes ... \otimes M\otimes...\otimes I$, where $M$ has index $i$ in the previous notation? I know that was not a very formal presentation, please let me know if the question cannot be understood.

If yes, can you please give an insight or maybe a more formal proof on WHY is it the case? Does it work even for entangled states?

Thank you very much for your response!


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