 Solution

Include all fixed resource states and all measurement records in an environment, so the entire physical protocol has an linear isometry $U$ from the unknown input to a larger pure output. If every recipient receives a perfect copy, its reduced density matrix is the pure projector $|\psi ⟩⟨\psi |$. A pure marginal forces factorization: write a joint vector in a basis beginning with $|\psi ⟩$; its squared components along every orthogonal basis vector vanish because their marginal probabilities are zero. Applying this successively to all recipients giveswhere $|R⟩$ is the fixed initial resource and $|E_{\psi }⟩$ may depend on the input. This includes Alice's discarded systems and every possible outcome record.

Choose two distinct nonorthogonal pure states, for example $|0⟩$ and $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, and write their overlap as $s$. Preservation of the inner product by the linear isometry requiresNormalized environment vectors have overlap of absolute value at most one. Thus $0<|s|<1$ would imply $|s|\le |s{|}^N$, impossible for $N>1$. Hence perfect transmission of a copy to every recipient is impossible for $N>1$. This proves the no-cloning theorem obstruction even though Alice keeps no copy herself.