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1

I think there are two nested questions there: How can you simulate partial measurement? There are a few approaches, but one basic one is described here. In short, zero the amplitudes of the states which didn't happen (i.e. ones for which the qubit you measured has the opposite value than the one you measured) and then normalize remaining amplitudes so that $...


3

The relation $U\otimes I=I\otimes U$ is wrong, unless $U=\alpha I$ for some $\alpha\in\mathbb C$. You have $$(U\otimes I)|\Phi\rangle =\sum_i (U\otimes I)|i,i\rangle = \sum_i (U|i\rangle)\otimes |i\rangle = \sum_{ij} U_{ji} (|j\rangle\otimes|i\rangle) \\= \sum_{j} |j\rangle\otimes \left(\sum_i U_{ji}|i\rangle\right) = \sum_j |j\rangle\otimes(U^T |j\rangle) =...


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