No-Existence Of Generalize Diffusion

We show that given two arbitrary states $\mathrm{\mid }\psi ⟩,\mathrm{\mid }\phi ⟩$ it is impossible to compute the transformation: $\mathrm{\mid }\psi ⟩\mathrm{\mid }\phi ⟩\mapsto \mathrm{\mid }\psi ⟩(\mathbb{I}-2\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid })\mathrm{\mid }\phi ⟩$ The contradiction of the existence of such operator follows by showing that using it, two players can compute the disjoints of their sets in a single round and $O\left(\sqrt{n}\right)$ communication complexity, which shown by Braverman to be impossible \cite{Braverman}.