erties of unitary operators, their complex- exponential representation, and, finally, the representation of unitary operators through rotations on the Bloch sphere. Hermitian conjugation and unitary matrices Given an operator A, whose matrix representation is characterized by the complex coef- ficients Aij, the Hermitian conjugate of A, called A+, is defined by the coefficients aji, corresponding to the complex-conjugate, transposed matrix. By definition, an operator or matrix is called unitary if it satisfies the property A+ A = I. Exponential representation of complex numbers A complex number z=a+ ib has modulus or length = It is pos- sible to represent a complex number in the form Appendix N where O = tan-1(b/a) is the argument of z. One may also write: z=z(cos0isin)=zei where ei = cos0+isinOis a complex number called an imaginary exponential. Such notation is justified if one considers the definition of the real exponential exp(x) = ex: Substituting x = i0 into Eq. ( = (N4) yields = cos0 + isin 0, where the two series corresponding to the real and imaginary parts of ei have been substituted with their exact function definitions cos O and sin A, respectively. Noteworthy are the two identities exp(i)) -1, exp(ir/2) Exponential operator Assume an operator A satisfying the property A2 = I. We then define the exponential operator exp o(iAO) according to exp(iA0)=cos(0)I+isin(0)A Note in Eq. (N8) that the real part of the exponential involves the operator I and the imaginary part involves the operator A. One proves the above development by substituting the argument x = iAO in the series expansion in Eq. (N4) and using the Pauli matrices, rotations, and unitary operators property A2= = I: = cos(a)I +isin(4)A (in the above, the two series expansions are recognized to correspond to the functions cos O and sin 0, respectively). Rotation operators We now have the mathematical tools to introduce a new class of unitary operators generated by the three Pauli matrices X, Y, Z, and called rotation operators. Such operators a