nds to a real number, not an operator). We have, thus, obtained the definitions of the two main statistical parameters characterizing any observable measurement, namely the mean and the variance of the observed physical quantity. We call the uncertainty of the observable A the measured standard deviation VAA2 = AA. Similar measurements can be performed with any other observable quantity B, with uncertainty . A fundamental result in quantum mechanics, known as the Heisenberg uncertainty principle, states that given two Hermitian operators A, B the product of their Quantum measurements uncertainties measured in the system state 1/>/ must satisfy: (17.62) where [A, B] = AB - BA is the commutator of A, B (see demonstration in Appendix O). In the particular case (called conjugate observables) where [A, B] = i, and where 1/>/ has a modulus of unity, we have (17.63) which represents the more well-known form of the uncertainty principle. The key con- clusion of this principle could be stated as follows: two independent measurements of noncommuting observables cannot both reach arbitrary accuracy. The product of the corresponding uncertainties has a nonzero lower bound given by the right-hand side of Eq. (17.62), which we may call E. Thus, a relatively accurate measurement of A, e.g., A = 1/N where N may be any large number, implies that the accuracy in the measure- ment of B cannot be better than Ne. Thus, if N 8, we have 0, corresponding to absolute accuracy in the observable A, while we have 80, corresponding to an absolute "indetermination" of the observable B. It would be incorrect, however, to conclude that in any case, a first measurement of A actually influences that of B, and the reverse. The two measurements are assumed to be independent, both in terms of time and order sequence. Only in the case where the two observables commute can the two measurements reach arbitrary accuracy. POVM measurements A third type of measurement is referred to as positive-operator-valued-measure, or POVM. Define a finite set of Herm