A Simple Proof of No Cloning

We provide a simple proof of the no cloning theorem from quantum computing.

11 December 2024

The no cloning theorem is traditionally stated in terms of unitary operators. In this post, we give a simple proof of a generalized version of this theorem.

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Theorem. The mapping Δ:V→V⊗V\Delta: V \to V \otimes V

v↦v⊗v\begin{equation} v \mapsto v \otimes v \end{equation}

of sending an element to the diagonal is not a linear transformation.

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Proof: Suppose otherwise, and consider the mapping on u+vu + v. We compute:

u⊗u+u⊗v+v⊗u+v⊗v=(u+v)⊗(u+v)=Δ(u+v)=Δ(u)+Δ(v)=u⊗u+v⊗v.\begin{equation} \begin{split} u \otimes u + u \otimes v + v \otimes u + v \otimes v &= (u + v) \otimes (u + v) \\ &= \Delta(u+v) \\ &= \Delta(u) + \Delta(v) \\ &= u \otimes u + v \otimes v. \end{split} \end{equation}

Moving terms around, we find u⊗v=−v⊗uu \otimes v = -v \otimes u, which is false for general vectors u,vu, v. □

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Note that we don't need to appeal to any norms or Cauchy-Schwarz, which means this theorem holds in more general categories than normally stated. It's also interesting to observe that if we were in the exterior algebra, then cloning would be allowed.