Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 3 / d / ii / Solution

Assume perfect cloning of both specified pure states. Allowing irrelevant output global phases, this would requireA unitary operator preserves inner products, and therefore their magnitudes. But the input overlap and required output overlap have different magnitudes:This contradiction proves that no such two-qubit unitary exists, without assuming the no-cloning theorem.