or linear super- position (q) = a/0)+ B|1), where |0), |1) form an orthonormal basis in the 2D vector space, and a, B are complex numbers, which represent the qubit coordinates in this space. The length of the qubit vector lq is, therefore, given by /a/2 + 1812. Since any two complex numbers a, B can be defined in the exponential representation as a = |aleie, and B = 1Blei82, one can write, from Eq. (M1): Assume next that the qubit vector is unitary, i.e., 1012+1812 = 1. Substituting this property and with the definition tan(0/2) = IB/A, we obtain from Eq. (M2): Finally, introducing y = A1 and Q=O2-01,the qubit takes the form lei== Overlooking the argument 8, which only represents an arbitrary (said "unobservable") phase shift, we finally obtain Appendix M Figure M1 Qubit represented as point of coordinates (A, 4) on Bloch sphere. The qubit can, thus, be represented through two angular coordinates A, , Q, which define the unique position of a point on a sphere of unit radius, which is the Bloch sphere illustrated in Fig. M1. It is seen from Fig. M1 that the pure qubits |0) or |1) correspond to O = 0 or O = IT, respectively, which occupy the north and south poles of the Bloch sphere. The qubit representation of quantum information, thus, corresponds to an infinity of states located on the surface of the Bloch sphere. See more on this topic in Appendix N. Appendix N (Chapter 16) Pauli matrices, rotations, and unitary operators In this appendix, I provide further conceptual background concerning Pauli matrices. It is shown that Pauli matrices make it possible to define any 2 x 2 unitary matrix U (namely, satisfying the property U+U=I, where the upper symbol (+) stands for Hermitian conjugation, and I is the identity matrix. The different results or theorems obtained in this appendix will be usefully applied to other chapters. Recall first the definitions of the four Pauli matrices: Note that the different notations 00,1,2,3, O0,x,y,z, and I, X, Y, Z are equivalent. We progress in several steps, which recall the prop