I guess it is *strange* to me that information about the system is partitioned so nicely. In particular, say I perfectly repeatedly prepare the same bipartite system $\mathcal{H}_1 \otimes \mathcal{H}_2$ in an entangled state $\lvert \psi \rangle$ in the lab.

Then, I take *local* measurements on subsystem 1 to compute an expectation value $\langle O \otimes \mathbb{I} \rangle$. That $\langle O \otimes \mathbb{I} \rangle$ models the measurement statistics I take in the lab seems strange. In particular, there is seemingly no composite system information leaking into the data collected by my …