bility 1 - p or p that using either filter 0 or 1 results in a measurement failure, and in these cases we do not get any classical bit information. There is one exception to this, however: if we know in advance that the qubit is in the superposition a = p = 1/ 2 = (case (b)), there are equal chances of success or failure, regardless of the measurement operator (or tone filter). There is no point in making any measurement, since we know for a fact that a successful outcome will give either a 0 or a 1 classical bit, with equal likelihood. We may just pick a bit value at random. The tax to pay for this result is one bit of advance information (i.e., "case (b)?" = YES, NO). The above description, despite its questionable usefulness from a purist or academic standpoint, made it possible to "visualize" somewhat and get a better feel of the nature of quantum measurements, when applied to single qubits. It is noteworthy, however, that the fictitious "magic-filter" experiment can actually be implemented in the physical world, as described in Chapter 25 in the part concerning quantum cryptography. The lesson learnt is that any "successful" qubit base-state measurement results in its collapse into one of the pure states, and the reduction of its intrinsic information into a classical 0 or 1 bit. In Chapter 17, we have also described another type of measurement referred to as projective or von Neumann measurement. In this approach, we can define the Hermitian observable A according to its spectral decomposition over the eigenstates 20), (21) Qubit measurements, coding, and teleportation forming an orthonormal base (18.5) where 20, A1 are the corresponding eigenvalues, and Pm = m) (m/ the measurement operators. Next, we recall the two key results obtained in Chapter 17, while applying them to the case of qubits. First, given a qubit lq = xol20) + - x1/21) as expressed in the eigenstate base V, the probability of measuring the eigenvalue Am is given by p(2m) = |xm|2. With the usual qubit notations |q)=a/20> + B121), we, thus, 