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This tag is for questions on the meaning, history, and usage of various symbols and notation that are used to denote different quantities in physics. Don't forget to mention the reference, where you encountered those notations.

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How does a unitary operator acts on an entangled $|\psi\rangle$?

Your unitary looks like a controlled-X from the second to the first qubit. It is fairly easy to work through what it does by just plugging in definitions. $U = |00\rangle\langle 00| + |11\rangle\la …
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