On Quantum Transmisisons, simplified.
"Untappable" quantum communications are based upon Heisenburg's Uncertainty Principle (read The Physical Principles of the Quantum Theory - Heisenberg 1930), and the theories of Bohr, Einstein, and Schrödinger.
The idea, in layman's terms, is that one cannot definitively measure quantum objects - any such measurements are inherently uncertain. Imagine this: a snooper wants to measure the relative position and wavelength of a photon (light particle) in a fiber-optic line, which represents one of the bits in your email transmission across the line. One must observe such a particle indirectly. To literally "see" a particle we measure - with our eyes or instuments - how photons are reflected back from the particle in question. By measuring their speed, position, and wavelength, we can infer the analagous qualities of our target.
The kicker comes when we realize that by observing a particle (particularly a quantumly-excited one), we necessarily change the characteristics of that particle. We can observe the speed, position, or wavelength of a particle by bouncing a photon off of it, but at the next instant, after the collision, the trajectory and wavelength of the particle are different, and the original information is LOST. (The we cannot ever know both the location and velocity of a particle). The detector at the other end of the fiber-optic line will notice that someone has attempted to observe the data en route (Man-in-the-Middle attack), and act accordingly - by sending another request using a new encryption key, etc. Of course, implementing such a system is currently limited to very short distances (meters, perhaps?).
Quantum computing works on other principles that I am less familiar with, but certain kinds of mathematical problems will be made much simpler with the quantum computers - problems which include prime factorization, a process which in integral to breaking many modern forms of encryption.
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