rlook this academic note, and abide by the above general definition of quantum- measurement operators, for which phase (and multiplying factors eis in quantum states) is indeed "unobservable." We shall consider next three possibilities for the measurement operator Mm. Quantum measurements in the orthonormal basis We look for a quantum measurement operator Mm that projects the input state 1/4/ into the pure state /xm) from the orthonormal basis V={|x)} = {|x1), /x2). , |xn)}, namely, having the action 1/4' = Mm/Y) = M/Xn), where u is a complex number. As we know, the projector over (xm) is defined as Pm = /xm) (xm), with P + Pm = Pm. The set of Hermitian operators {Pm = /xm) (xml} satisfies the completeness relation, hence Mm = (xm) <xml is a valid measurement operator. We first derive the bracket (V|M+MM/Y) according to -(Exal) (17.52) i=1 j=1 which shows that the probability of measuring m (or equivalently, the probability that 1/4/ is in the state /xm)) is p(m) = |xm12. Then, with substitution of Mm = 1xm) <xml into See, for background reference: E. Desurvire, Erbium-Doped Fiber Amplifiers, Device and System Devel- opments (New York: J. Wiley & Sons, 2002), pp. 97-111. As an advanced topic for quantum-mechanics specialists, attention is drawn to phase states 18m), which have eigenvalues 8m, and which are defined for electromagnetic fields by where |k> is a photon-number state, and the series applies in the limit S 00. For full description, see: S. .M. Barnett and D. T. Pegg, On the Hermitian phase operator. J. Modern Optics, 36 (1989), 7-19 and D. T. Pegg and S.M. Barnett, Phase properties of the quantized in single-mode electromagnetic field. Phys. Rev. A, 39 (1989), 1665. Quantum measurements Eqs. (17.51) and (17.52), we obtain (17.53) The result in Eq. (17.53) indicates that the post-measurement state is indeed of the form u/Xm), but with u = xm/lxml being a complex number of unity modulus, or u = eism being an "unobservable" phase term that can be overlooked. We will use the above results in Chapter 18, whi