 the qubit lq>. In vector notation, Eq. (17.3) can be written as a line-vector-column-vector product: (17.4) where [q)T is the conjugate-transposed of lq). Here is a first opportunity to show how Dirac notation comes in handy. Indeed, define the "bra" <ql as representing the 17.1 Dirac notation conjugate-transposed vector 1g)T With this notation, Eq. (17.4) becomes the "bra-ket:" (919') = ' + BB' (17.5) And, hence, for the qubit modulus, As another property, we note from Eq. (17.5) that namely that the inner product of qubit states is noncommutative, unlike the scalar product of 2D vectors with real coordinates. As we shall see next, there is more in the "bra" notation than just a convenient way to express the inner product between two qubit states. Indeed, consider n-qubits /q), |q'> in the space Vn, with, for |q'), the expansion in Eq. (17.2) and, for the bra, (q| (consistently with definition): (17.6) We can now develop the inner product (q|q') according to (17.7) Since |x), |x') are pure states from the orthonormal basis Vn, we have (x/x') = 8ss' where 8ss' is the Kronecker symbol (8ss = 1 for X = x' and 8ss = 0 otherwise). Hence the explicit definition of inner product when the two qubits are expressed in the same basis Vn: (17.7) XEV"X'EV" Which, for reduces to the modulus: (17.8) In particular, effecting a left product by (x| to Eq. (17.2) we have (17.9) which shows that, expectedly, (x/q) = x is the projection of lq) over the pure state /x). Substituting this result into the definition in Eq. (17.2), we obtain (17.10) xEVnxEVnxEVn which represents another way of expanding lq) over the basis Vn. In Chapter 16, we have shown that it is generally not possible to "clone" quantum states, a property referred to as the noncloning theorem. As was mentioned without Quantum measurements proof, the only exceptions to this theorem concern any pure state, or any pair of pure states. With the inner product introduced above, it is quite easy to show this. 1 Projection operators In view of the expansion of lq) in Eq. (17