In classical physics, light of a single color is described by an electromagnetic field in which electric and magnetic fields oscillate at a frequency, that is related to the wavelength, by the relation c = , where c is the velocity of light. For example, visible light has wavelengths in the range from 400-750 nm, while longer wavelength radiation (invisible to the eye) is known as infra- red.
Polarized Optical Waves
An important property of optical waves is their polarization: we will define a vertically polarized ("V") wave as one for which the electric field is restricted to lie along the z-axis for a wave propagating along the x-axis, and similarly a horizontally polarized ("H") wave will be defined as one in which the electric field lies along the y-axis. It is a remarkable property of light that any other polarization state of light propagating along the x-axis can be resolved into a linear superposition of vertically polarized and horizontally polarized waves with a particular relative phase. In the case of linearly polarized light the amplitude of the two components is determined by the projections of the polarization direction along the V or H polarization axes. For instance, light linearly polarized along the +45† direction in the y-z plane is an equal amplitude, in-phase superposition of V and H, while light polarized along the -45° direction in the y-z plane is an equal amplitude, opposite-phase superposition. For obvious reasons V and H polarization states (or +45° and -45° polarizations) will be referred to as orthogonal polarizations, while two polarizations (such as V and +45°) that have a non-zero projection will be called non-orthogonal .
Light of a particular linear polarization can be produced by sending unpolarized light through a polarizing medium (Polaroid) whose polarizing axis is oriented along the direction of the desired linear polarization. When this light is passed through a second polarizer, only the component polarized parallel to the polarizing axis emerges, while the orthogonal component is absorbed. For example if V light impinges on a polarizer oriented at +45° the emerging light is reduced in amplitude by a factor of 2^-1/2 (one over the square root of two), has the +45° polarization and an intensity (proportional to the square of the amplitude) which is 50% of the incident intensity. Likewise, if V light impinges on an H polarizer, no light emerges, and we refer to this configuration as having "crossed polarizers."
Foundations of Information Science
These are the essential features of classical polarized light, but in quantum cryptography we deal with very low intensity light where quantum mechanics must be used. Specifically, during propagation such light has wave-like properties, but on detection exhibits particle-like behavior: the optical energy is quantized into indivisible units of size h, called photons (the elementary particles of electromagnetic radiation) where h is a fundamental constant of Nature known as Planck's constant. The indivisibility of photons raises the interesting question of how a photon of polarized light behaves when it encounters a polarizer. Clearly, there is no difficulty of interpretation if we are only concerned with orthogonal polarizations: a V-photon would pass a V-polarizer with certainty, but be absorbed by an H-polarizer with certainty, for instance. However, once non-orthogonal polarizations are introduced the peculiarities of quantum mechanics become evident.
Consider a +45°-polarized photon impinging on a V-polarizer: there is no such object as a "half photon" by analogy with the classical 50% transmission intensity.
Collapsing the Photon's Wave Function
Instead, quantum mechanics predicts that there is a 50% probability that the photon will be absorbed, and a 50% probability that it will be transmitted, with V-polarization, in any given trial (experiment). Of particular relevance for quantum cryptography is that beyond these probabilities we cannot predict how a particular +45°-photon will behave at a V-polarizer. (Of course the result of a given experiment is either a definite absorption or transmission, but with many repetitions of the experiment we would build up a set of results reflecting the 50-50 absorption-transmission probabilities.) A further relevant quantum peculiarity is that if a +45°-photon passes the V- polarizer it loses all of its "+45°-ness." Specifically, if the emerging photon is made to impinge on a second polarizer oriented at +45° it will be absorbed or transmitted with 50% probability in each case, even though it was originally a +45°-photon. This randomization of properties by non- orthogonal measurements is a crucial element in the detectability of eavesdropping in quantum cryptography. In the terminology of quantum mechanics one says that the V-polarizer has "collapsed the photon's wavefunction."
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