 M/O)) (2/2/5) + M/O)) (16.63) 22/4, + MA/V, + MA/O, v) Equating Eqs. (16.62) and (16.63) yields 1)1,4,4 + MA/V, 0) + M2/O, y/s M(u 1),10,00 = 0. (16.64) Assuming that 1/s/, |0) are pure states, the above equation implies that un = 0 and, thus, 1x) = /V/ or 1x)=00. This result means that if there exists an operator U that can clone two pure states 14), |0), this operator cannot clone any of their mixtures 1x)=2/4> + MO), which is a quite restrictive conclusion. We are then left with the open question: does any cloning operator U exist in the first place? The answer is straightforward, but it requires one to use the inner product of states, which is introduced in Chapter 17. Suffice it here to provide the result: there always exist a unitary operator U capable of cloning a pure state 1/4/, or any pair of pure states 1/5/ and 15) (see Chapter 17 for proof). As we have previously seen, however, such an operator cannot clone the mixture 1x) = The key conclusion is that, except for the limiting case of pure-state pairs, it is not possible to clone quantum states in the general case. This fundamental result is known as the noncloning theorem. In the specific case of qubits, the two possible bases of pure states are {|s1>, |s2)} = With our knowledge of quantum gates, it is trivial that we can find operators capable of "cloning" pure states into each other. Thus the exception about 16.7 Exercises pure states does not weaken in any sense the generality of the noncloning theorem. In particular, there is no quantum gate capable of executing the equivalent of the classical FANOUT gate (Chapter 15), for any state other than a pure state. Exercises 16.1 (B): Show by two different methods that the Hadamard gate H corresponds to a unitary transformation. 16.2 (T): Prove the property according to which the three operators Rk(s) (k = = X, y, z) rotate any qubit |q) on the Bloch sphere by an angle y about the axis k, in the counterclockwise direction. 16.3 (B): Prove the following properties of the rotation operators: R;(20)R;(-20)