 the values +1 or -1, respectively, to these two possible measurements. Recall from Chapter 16 that |0), |1), and +1 are the eigenvectors and eigenvalues of the Pauli Classical and quantum cryptography Figure 25.5 Splitting a light beam of single, linearly polarized photons through a polarization beamsplitter (PBS). The apparatus corresponds to a measurement of photon states in the Z basis. Figure 25.6 Splitting a light beam of single, circularly polarized photons through a polarization beamsplitter (PBS) preceded by a quarter-wave plate (QWP). The apparatus corresponds to a measurement of photon states in the X basis. matrix Z, according to: (25.22) Thus, measuring linearly polarized photons is equivalent to "observing" Z, or equiv- alently, to operating in the measurement basis {|0), |1)}, or, for short, in the Z basis. A different set-up, which involves circularly polarized photons, is shown in Fig. 25.6. The generation of left- or right-circularly polarized photons is achieved by plac- ing a QWP (quarter-wave plate) next to a linearly polarized SP source, with its fast axis at 45° CW or ccw, respectively. A way to tell whether a photon is left- or right- circularly polarized is to convert its polarization into a linear polarization, followed by the 25.8 Electromagnetic waves, polarization states, photons, and quantum measurements Table 25.7 Probabilities p(#1) of measuring eigenvalues 1 (counts in detectors SPD1 or SPD according to input photon state (I, <->, n, |0), |1), 1+), |-)) and measurement apparatus (X, Z). Input photon state I or |0) or |1) or 1+) U or |-> Measurement basis p(+1) p(-1) PBS-SPD1-SPD2 apparatus, as illustrated in the figure. The QWP-PBS-SPD1-SPD2 set-up, thus, constitutes a quantum measurement apparatus to determine the state of circularly polarized photons. If a count is obtained from SPD1 or from SPD2, we may attribute the values +1 or -1, respectively, to these two possible measurements. Recall from Chapter 16 that 1+), |-), and +1 are the eigenvectors and eigenvalues of the Pauli m