alling from Chapter 16 that the Hadamard matrix, H, precisely achieves the four transformations H |0) |+) |1) = |-> (25.21) H +)=|0) = |1). This observation establishes that a QWP device transforms the polarized EM states (1, <>, , n, U) in a way strictly equivalent to that of the Hadamard matrix, H, on the quantum states (|0), |1) ++), 1-)). From the above description, we have seen that it is possible to manipulate the polar- ization states of the EM field. To progress further, we must, at this stage, take into account the quantum nature of light, namely the particle-like behavior of the photon. Such behavior can be readily understood by considering the two experiments illustrated in Figs. 25.3 and 25.4. In the first experiment (Fig. 25.3), we divide an incident EM light beam with E-field E into two beams of equal E-field amplitudes; this is achieved through a 50:50 beamsplitter. Since the incident EM power is given by P = |E|, the E-field amplitudes of the two output beams are E/V2. Two detectors placed on the Classical and quantum cryptography 50 : 50 Beamsplitter Light source Detector Detector Figure 25.3 Dividing an EM light beam (electrical field E, power P = |E| through a 50:50 beamsplitter, resulting in two beams (electrical fields E/2) of detected powers both equal to P' = P/2. 50 : 50 Beamsplitter SP source Detector Detector Figure 25.4 Same experimental apparatus as Fig. 25.3 with single photons (SP) being emitted and detected. output beam paths measure equal EM powers of P' = P/2, as expected. The picture becomes very different if we reduce the EM power to the point of reaching the level of photon granularity, i.e., the number of photons emitted per second by the light source is of the order of unity. The experimental apparatus shown in Fig. 25.4 is basically identical to the previous one, except that now the light source produces single photons1 17 (SP) and that the two detectors are capable of detecting such single photons. Since photons are particle-like energy quanta, they cannot be split. On reac