Proof of no cloning theorem

I’m still studying the online course on quantum computing where Prof. Vazirani went through the no cloning theorem. It is very interesting to see that something that can lead to some very subtle consequence can also have a very simple proof. Below is a summary.

Basically, the theorem states that no quantum state can be cloned by quantum evolution, the only allowed operation in reality. Let say we want to clone a quantum state |\psi\rangle=a|0\rangle+b|1\rangle by passing the product state |\psi\rangle \otimes |0\rangle through some unitary operator. The desired output should then be |\psi\rangle \otimes |\psi\rangle=a^2|0\rangle+ab|01\rangle+ab|10\rangle+b^2|1\rangle. In particular, if we consider the cases when |\psi\rangle=|0\rangle and |\psi\rangle=|1\rangle, the outputs should be |00\rangle and |11\rangle, respectively. However, by linearity, this implies that the output when |\psi\rangle=a|0\rangle+b|1\rangle should be a|00\rangle+b|11\rangle instead. Therefore, cloning is possible only when ab=0, i.e. when |\psi\rangle is either |0\rangle or |1\rangle.

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