ch concerns qubit measurements in the orthonormal basis. Projective or von-Neumann measurements Another type of quantum measurement whose operator satisfies the above two postulates is referred to as projective or von-Neumann measurement. As we shall see, it is not formally different from a measurement in an orthonormal basis, except that we shall now take for this basis the one formed by the eigenstates {12m}} of a given Hermitian operator A. Define the corresponding eigenvalues as {2m}. Since A is Hermitian, we know from the previous section that the eigenvalues are real numbers. For this reason, the operator A could be associated with "real" physical quantities to characterize the system, hence the operator A is referred to as an observable. Given the set of projectors Pm = 12m) (Aml, the spectral decomposition of A is given in Eq. (17.27), which I reproduce for convenience: (17.54) As we have previously established, the projector Pm n=12m) (2ml is a valid measurement operator and, hence, the probability of measuring the eigenvalue Am from a system in the state 14), as decomposed in the eigenstates basis {12m)} is given by (17.55) and we have for the post-measurement state: (17.56) 17.2 Quantum measurements and types Substituting the expansion of 1/s/, with complex coordinates Xi, over the eigenstate basis {|2i}}, we obtain for the measurement probability p(2m) and corresponding post- measurement state 14'): (17.57) (17.58) We will use the above two results in Chapter 18, which concerns projective qubit measurements. Let us now analyze some interesting properties of the projective or von-Neumann measurement. Since each eigenvalue Am has an associated measurement probability, p(am), we can define the average value or mean of the observable measurement as follows: (17.59) The nice conclusion from the above result is that the average value of an observable A is defined as the bracket (VAV). Likewise, we can define the mean-square and variance of the observable as follows: (17.60) (17.61) (the notation AA correspo