late classical ones, with qubits instead of bits! (Chapter 19 and 20 will prove far differently). Tensor products In previous sections, we made use of the notion of tensor product for qubits, to describe quantum gates with more than one qubit input. This section will make it possible to clarify such a notion, which is fundamental to the understanding of quantum computing with multiple states. In the description, we shall proceed gradually, from elementary to advanced. Given two qubits |a), |b) we called the tensor product the 2-qubit |a>|b). From now on, we shall write the result with the new notation with the sign X standing for the tensor-product operation. Given the computational base V = {|0},|1)}, which defined a 2D qubit space, we can form the 4D extended base defining a 4D 2-qubit space V X Vaccording to: VQV{0),(1))0{0),1)} = |1)|1>} (16.45) = {10)0(0),(0)01),1)00),1)01 In the extended base, |a) 10/b/ is represented by four complex coordinates (u1, U2, U3, u4) to be determined. Assuming that |a) = aol0) + a1|1) and |b) = bolo) + b1|1), we obtain |a) bl = =(ao0)+a11))0(b010)+ b1|1)) (16.46) = x |1), hence (u1, U2, U3, u4) = In the above, we have implicitly made use of the linearity and distribution-over-addition properties of the tensor product: (16.47) The notion of tensor product, along with the same properties, applies to any two nD and mD spaces defined by computational basis V ={|a1> la2) |an)} and W = {|B1>, 1B2> 1Bm)}. Hence, given (16.48) we have (16.49) A qubit |a) that is tensored with itself n times will be noted |a) namely, |a)=a)a),...,a)ntimes. (16.50) Quantum bits and quantum gates The tensor state |a) x |b), often noted |a, b) for simplification, is also referred to as a joint state. Such a joint state corresponds to the description of two quantum systems, themselves being in the states |a) and |b), respectively. Thus, |a) x |a) corresponds to the description of two systems being in the same state |a). However, such a possibility should not lead one to the conclusion that given a quantum sy