The no-cloning theorem states that it is impossible to perfectly copy an unknown quantum state. That is, given an unknown [Pure state](Pure%20state.md) $|\psi\rangle,$ we cannot construct a [unitary operator](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics) $U,$ such that $\hat{U}(|\psi\rangle\otimes|\phi\rangle)=|\psi\rangle\otimes|\psi\rangle$ where $|\phi\rangle$ is a second pure state we desire to make [identical](identical%20particles) to $|\psi\rangle.$ The final output of such an operation would be a [separable composite state](Separable%20states.md) $|\psi\rangle\otimes|\psi\rangle,$ containing two copies of $|\psi\rangle.$ %%Include here also a reference to the same theory presented in terms of quantum gates - that should be from Nielsen and Chuang%% --- # Proofs and examples ## Proof of the No-cloning theorem    --- # Recommended reading For the original proof of the [No-cloning theorem](No-cloning%20theorem.md) see: * [Wooters, W. K., Zurek W. H., _A single quantum cannot be cloned_, Nature, 299, 1982](Wooters,%20W.%20K.,%20Zurek%20W.%20H.,%20A%20single%20quantum%20cannot%20be%20cloned,%20Nature,%20299,%201982.md) Here the no-cloning theorem is described and proven by considering a pair of photon polarization states. %%Relevant ideas connecting this note: Unitary time evolution, (time evolution operator), hilbert space tensor product? Composite quantum state? Yes we need a note for composite quantum states.%% #QuantumMechanics/QuantumInformation #QuantumMechanics/FoundationsOfQuantumMechanics #QuantumMechanics/MathematicalFoundations/QuantumInformation