1) + ax'|10) - aB'|01))34 = - a' |||||||||||||||||||||||||||||||| + aB'|10) - aa'|01))34 = + aB'|11) - - BB'|01))34 = + aa'|11) - BB'|10) - a' B|01))34 = - aB'|11) - x' B|10) + BB'|01))34 = - aa'|11) - BB'|10) + -a' B|01))34 = ZZZZXXXXX(-a/B|00) - BB'|11) + aa'|10) + aB'|01))34 = - + aB'|10) + aa'|01))34 = + BB'|11) + aa'|10) - aB'|01))34 = x' |||||||||||||||||||||||||||||||||||||||||||||||||||||| + aB'|10) - aa'|01))34 = 1/>/. (P17) Two-qubit teleportation The first equality in Eq. (P17), corresponding to \mnm'n'> = |0000) is immediately established by comparison with Eq. (P16). All the other equalities can be verified with the understanding that they hold within an occasional (unobservable) phase factor eir = - -1. As an illustration of the verification, consider, for instance, the last equality in Eq. (P16), and apply the Pauli operators as defined: Umnm'n'10)34 = - = + a' B|00) + -aB'|01) - aa'|10))34 = Z}Z}(BB'|01) + a / B|10) - aB'|11) - aa'|00))34 = - - = (-BB'|01) - a' |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| - aB'|11) - aa'|00))34 = -(aa'|00) + aB'|11) + -BB'|01))34 (P18) The operation of the quantum circuit, together with Alice transmitting her four measured cbits m, n, m', n' to Bob, thus, result in the successful teleportation of the tensor state as defined in Eq. (P16). As mentioned at the beginning of this appendix, Bob can retrieve Alice's qubits 19), |q') by effecting a final CNOT operation on 1/4/, as described by the operator C34. Indeed, 1 Yout) = = C43/Y'Y = (aa'|00) + aB'|01) + + BB'|11))34 (P19) which shows that 1/4' is the unique tensor product of the qubits Appendix Q (Chapter 19) Quantum Fourier transform circuit In this appendix, I develop a simplified tensor-product expression for the components resulting from the quantum Fourier transform (QFT) of N-qubits. This expression leads to the conception of a corresponding QFT circuit based on simple 2 X 2 quantum gates. The starting point is the fundamental QFT definition (Eq. (19.16))