Abstract. When an electromagnetic signal propagates in vacuo, a polarization detector cannot be rigorously perpendicular to the wave vector because of diffraction effects. The vacuum behaves as a noisy channel, even if the detectors are perfect. The "noise" can however be reduced and nearly cancelled by a relative motion of the observer toward the source. The standard definition of a reduced density matrix fails for photon polarization, because the transversality condition behaves like a superselection rule. We can however define an effective reduced density matrix which corresponds to a restricted class of positive operator-valued measures. There are no pure photon qubits, and no exactly orthogonal qubit states.
IntroductionThe long range propagation of polarized photons is an essential tool of quantum cryptography [1]. Usually, optical fibers are used, and the photons may be absorbed or depolarized due to imperfections. In some cases, such as communication with space stations, the photons must propagate in vacuo [2]. The beam then has a finite diffraction angle of order λ/a, where a is the aperture size, and new deleterious effects appear. In particular a polarization detector cannot be rigorously perpendicular to the wave vector, and the transmission is never faithful, even with perfect detectors. Moreover, the "vacuum noise" depends on the relative motion of the observer with respect to the source.The relativistic effects reported here are essentially different from those for massive particles [3] because massless particles have only two linearly independent polarization states. The properties that we discuss are kinematical, not dynamical. At the statistical level, it is not even necessary to involve quantum electrodynamics. Most formulas can be derived by elementary classical methods as shown below. It is only when we need to consider individual photons, for cryptographic applications, that quantum theory becomes essential.This article consists of two parts. First we consider the propagation of a classical electromagnetic wave. The wave vector cannot be constant because of diffraction effects. A polarization detector cannot unambiguously distinguish orthogonal polarizations, even if the detector is perfect. The vacuum behaves as a noisy channel. We then show that this "noise" can be reduced and nearly cancelled by a relative motion of the observer toward the source.In the second part of this paper, we investigate the transmission of a single photon. The