Wikipedia states that a controlled U gate maps the basis states to the following:

$$ |00\rangle \mapsto |00\rangle\\ |01\rangle \mapsto |01\rangle\\ |10\rangle \mapsto |1 \rangle \otimes U |0 \rangle = |1 \rangle \otimes \left(u_{00}|0 \rangle + u_{10}|1 \rangle\right)\\ |11\rangle \mapsto |1 \rangle \otimes U |1 \rangle = |1 \rangle \otimes \left(u_{01}|0 \rangle + x_{11} | 1 \rangle\right)$$

I am wondering how was this mapping derived?

I have looked at Nielsen and Chuang - Quantum Computation and Information 10$^{th}$; but I wasn't able to find the derivation process.



It is not a derived mapping, it's the mathematical version of the definition of how a controlled gate works: it does nothing to the second qubit if the first qubit is $|0\rangle$, and applies the unitary $U$ to the second qubit if the first qubit is $|1\rangle$.

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  • $\begingroup$ Thank you for the explanation :) $\endgroup$ – M. Al Jumaily Aug 17 '19 at 18:21

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