Which is in contradiction with our initial assumption to be able to copy any arbitrary state.

Thus the prohibition against cloning helps preserve privacy.

As discussed earlier, if her cloning device can success-fully copy the vertical and horizontal polarizations, it will fail to copy faithfully either of the two diagonal polarizations. This result in the form of X = X 2 leads to the conclusion that we can copy the state |Ψ 1> and |Ψ 2> only if =0 or =1, on other words only and only if the states |Ψ 1> and |Ψ 2> are either parallel or orthogonal. because of the no-cloning theorem, Eve cannot succeed in her task. Taking the inner product of these two expressions we get:Īs the C operator is unitary, by definition CC*=I (identity matrix) so that Since the theorem is about the ability of C to clone any arbitrary state, let’s write down the copy for two random states |Ψ 1> and |Ψ 2> that should be copied into the state |ρ> That is C is a two-qubit operator that copies or clones the first qubit into the second qubit And so we are left with the paradox of either the cloning of information. linearity of quantum mechanics, to show that no apparatus, however complicated, can amplify. Suppose that there exists a unitary operator C that copies an arbitrary unknown q-state. For a more detailed explanation of the no-cloning theorem, please see Appendix A. ity of cloning photons, shown below, thus prohibits super. Theorem: It is impossible to create an identical copy of an arbitrary unknown quantum state. Formally, it states that we cannot clone arbitrary quantum states by unitary evolution. Why don’t we just make a copy of it? Unfortunately this is what the no-cloning theorem forbids. She can not measure the qubit to see what it was, for that this would inherently modify the qubit by collapsing its state. Suppose that Alice has some unknown qubit that she wants to send to Bob.

No clone theorem