.10), we can introduce Ux=|x)(x) as the operator projecting any qubit onto the state /x). To show this, we first observe that (17.11) (17.12) xx)x)x)xxxx), x' EV n which shows that /x) is invariant by Ux and also that lq) is projected on its basis component x/x). With such a definition of projector operator, we also have from Eq. (17.10): (17.13) from which we obtain (17.14) The above property, which is referred to as the completeness (or closure) relation, expresses the fact that the complete sum of all projection operators over the pure states of Vn is the identity operator. More generally, a unitary operator U is a projector if it satisfies the property U2=U (projecting twice is the same as projecting once). In the case of have, indeed, (17.15) =x)(xx)(x=x)dxx(x/=x)x=U Referring to Section 16.6, given three pure states |s), 1/4/, 10), we assume a unitary operator U capable to achieve the following cloning transformations: U(0)0s))=0)00). Taking the inner product of the two right-hand sides and using the properties U+U = I and (s/s) = 1, yields: which shows that either (0/V) = 1 or (0/V) : 0, corresponding to (0)==>) or |0), 1/4/ being orthogonal states. 17.1 Dirac notation (17.16) These two results confirm that Ux=|x)(x) is the unique projector on the pure state /x). It is clear that for any state lq) of unity length (such as a qubit), the operator 1q) (q| is the projector over the state /q), which is left as an easy exercise to demonstrate. Change of basis Let Wn = {|y)} be a new orthonormal basis. Consistently, we can expand |q) over Wn, using a new system of coordinates y defined over the states ly). How can we relate the new coordinates y to the old coordinates x? Here again, Dirac notations turn out to be quite handy. First, let us insert the completeness relation of Wn into the Vn expansion according: (17.17) The result in Eq. (17.17), thus, yields the definition of the new y coordinates with respect to the old x coordinates: (17.18) Unitary transformations As seen in Chapter 16, a unitary transformation