 means that there is a 2/3 chance that this measurement fails. Likewise, there is a 1/3 chance that the measurement of 1 fails when using M1. But what does "measurement failure" mean? Basically, that whenever we choose to use a given measurement projector Mm, the input state is not absolutely certain of collapsing into the corresponding state /xm). It will only happen "successfully" with probability |xml2. If this collapse does not happen (with probability 1 - |xml2, then nothing is measured, and the measurement is, thus, failed. Also, the quantum state col- lapses into nothingness, which here means qubit annihilation and irreversible loss of its information. We can mathematically define such an annihilated, informationless state as |q') = . To clarify the picture, or to alleviate some possible unease with the notions of "suc- cessful" and "failed" qubit measurements, consider the following (fictitious) physical analogy, which we have already used in Chapter 17 for POVM measurements. Compare the qubit lq) with a source that has two possible tones, A or B (standing for the states |0) or |1), respectively), and which are not visible to the eye. Our measurement consists of determining the color tones by observing this source through a set of two "magic filters," called 0, 1 (for M0, M1). As we have seen earlier, such filters have strange properties. Indeed, filter 0 does not react to tone B, while it makes tone A visible to the eye, but only with probability p (or /a/2. Filter 1 does not react to tone A, while it makes tone B visible to the eye, but only with probability p (or 1B12). Even stranger here (than the description in Chapter 17) is that the two filters A, B are of complementary nature, since their probabilities of positive reaction are now complementary (q = 1 - p). Figure 18.1 shows the eye images obtained in the eight possible cases, according to the two pure-tone possibilities A or B (but not both) and their mixture A, B, and the two filter options 0, 1. The observation of a bright spot indicates a posi