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I'm really confused with the interpretation of those equations:

$1.$ The evolution of states under unitary operations can be expressed as $$ U = \sum_k\exp(i\phi_k)|\psi_k\rangle\langle\psi_k| $$

$2.$ The controlled operation in quantum computing is defined as $$ |0\rangle\langle0|\otimes I + |1\rangle\langle1|\otimes e^{i\alpha}I = \begin{bmatrix} 1 & 0\\ 0 & e^{i\alpha} \end{bmatrix} \otimes I $$

I'm not pretty sure how to understand them. Also, the outer product is present in both equations, so I'm wondering could those equations be explained in terms of the projection operator? What are the physical meanings of the outer product?

Thanks:)

Please link to the source where you found this. – keisuke.akira – 2020-11-01T05:30:19.797

@keisuke.akira Thanks for the answer, the link I found this is https://www.youtube.com/watch?v=iLcQ-X6QzvU

– Zhengrong – 2020-11-01T14:31:57.690