asurements, which we index m (m = 1,..., n). We then introduce the most general definition of a measurement operator, Mm, the collection of which forms the finite operator set {Mm}. Calling p(m) Quantum measurements the probability of measuring m, we have by postulate: (17.46) Since the probabilities must sum to one, we also have (17.47) which implies the "completeness" relation: (17.48) As a second postulate, the measurement Mm causes the system state 1/s/ to be trans- formed into the post-measurement state 1/4') = y Mm/V), where y is a complex number. The probability q that the system is in the post-measurement state 1/4'' is unity. We, thus, obtain (17.49) = = 1, which shows that 121 = 1/1p(m), hence y = Within an "unob- servable" phase term eis, which we shall overlook here, we can set y = 1/p(m), and the post-measurement state is now completely defined as: (17.50) Using the expansion in Eq. (17.34) of the input state 1/4/, we obtain the expansion of the post-measurement state 1/4'' according to (17.51) Mm/xi). We have, thus, obtained a general expression for the post-measurement state 14'), given the quantum measurement operator Mm. What could be the result of two successive measurements from different operators Mm and L1? The answer is that 17.2 Quantum measurements and types the two successive measurements (in this order) are equivalent to a single mea- surement of operator definition KIm = LMm, which is left as an exercise to demonstrate. An important consequence of the above general definition for the quantum- measurement operator Mm is that two states 1/5/ and which differ by the phase factor eis, have the same measurement properties and outcome. Hence such a phase factor is said to be "unobservable." It should not be concluded, however, that in quan- tum mechanics, phase is definitely unobservable, with no corresponding measurement operator. Indeed, more recent (1989) and less known work has shown that Hermitian phase-measurement operators and phase eigenstates can truly be defined.5 Here, we shall ove