 system state /v/s has been prepared in such a way that it is certain to be either 1/4/ = |0) or 1/4/ = 1+), to the exclusion of any other possibility. It is also assumed that we have this key information about the two system-state possibilities, prior to any measurement, but we don't know, a priori, which ones the system chooses to be in. The probabilities associated with our three possible measurements associated to each case are summarized in Table 17.1. First, the table indicates that our measurement fails to convey any information in two cases: If we use E1 when the input is /V/> = |0); If we use E2 when the input is This is because, in each case, the measurement operators project the input state on its orthogonal counterpart, namely |1) or |-), respectively, resulting in the measurement probabilities for E1 of p(E1) = (0|E1|0) = 0 or p(E2) = (+|E2|+) = 0, and showing that these measurement have no possible outcomes. Second, the table indicates that there is a finite probability u/2 that we obtain a measurement Of E1, if we use E when the input is = |+); Of E2, if we use E2 when the input is (v) = |0). Then comes a nice subtlety in the interpretation of these two measurement possibilities. Indeed, if we happen to measure E the system cannot be in state 1/5/ : |0), since E1 is Quantum measurements a projector on |1). In this case, we reach the absolute conclusion that the system must have been in state 1/4/> = 1+). Likewise, if we happen to measure E2, the system cannot be in state = 1+), since E2 is a projector on |->. In this case, we reach the absolute conclusion that the system must have been in state 1/4/ = |0). However, we obtain this absolute information only with probability u/2, meaning that there is a finite probability 1 - u/2 that the two measurements fail to convey any information. If we were to use the measurement E3, Table 17.1 shows that the measurement works in all cases (1/v) = |0) or 1+)), but with a probability 1 - u/2. By definition, E3 is a projector on all states that are neither |1) or