B+A+, respectively. 16.5 Tensor products which enables one to calculate the tensor product |a) x |b). Let us next examine some illustrative examples. First, consider the case of the qubit tensor product |a) x |b) in bases V = {|a1), |a2>} and W = The corresponding matrices are single column, with Ai = ai and B1 = bi. We obtain: (16.55) which is the expected result. Second, consider the tensor product X X Y of the two Pauli matrices X, Y. We obtain: (16.56) It is left as an easy exercise to verify the property X X Y(|a) x |b)) = X|a) x Y|b), which, as we have seen, applies to any pairs of operators A, B and qubits |a), lb). An interesting case of n-tensored operator is provided by Hon where H is the Hadamard gate. Assume the extended computational base Vn = {|0), |1)}" = {|a)}, with |a) symbolizing any of the n-qubits base element generated by the n-tensor product |a) = /V1) X /Un) where Vi = 0 or Vi = 1 (i = 1,...,n). As a general property, it can be shown that (16.57) where @.../wn) is any base element of Vn , and a*b is a scalar defined as: (16.58) With the tensor-product tools developed in this section, it is proposed as a closing exercise to establish the property in Eq. (16.57) for the case n = 2, then by induction for the general case. Quantum bits and quantum gates Noncloning theorem Given a quantum system A, in any state 14), and a second quantum system B, in any pure state |s), is it possible to duplicate the first state into the second? Such a "cloning" operation would correspond to the transformation: 1/4/0/s) (16.59) where U is a unitary tensor operator. Assume that such an operator U exists and applies to any state /V/> of A. Let |0) be another state of A such that |0) # lv>. We must also be able to duplicate it into B according to: U(0)0s))0)0=0,0) (16.60) By linearity of the transformation, we must also have for any state mixture 1x) of A, where A, u are two complex numbers: (16.61) If we develop the left-hand side of Eq. (16.61) we obtain: (16.62) while the right-hand side in Eq. (16.61) yields: +