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Quantum Cryptography

What is commonly called "Quantum Cryptography" is a method that exploits the EPR (Einstein-Podolsky-Rosen) paradox to distribute a one-time-pad to two locations. It is secure against undetectable eavesdropping, but although an active attack could be detected, it could still lead to the attacker reading a message.

In quantum mechanics, a particle, instead of actually being a point object that can be in only one place at a time, is really a wave function that is spread over a larger or smaller area. The act of observing where a particle is consists of changing the shape of its wave function to one with a sharp spike at one location.

If a particle's wave function spans a large area, and quick measurements of position are made at the same time at two widely separated points in that area, the wave function, whose amplitude determines the particle's probability of being located at each point, controls two independent chances of the particle being where it is looked for, because nothing can travel faster than light. So you have a chance of looking to see where a particle is, and finding two of them. This led to the theoretical prediction of the existence of antimatter; in this way, an extra particle can be created, and there is no problem, because to make two position measurements so precisely and so quickly actually takes enough energy to create an extra particle, and its antiparticle as well (required to conserve various physical quantities).

But for some physical properties, one of which is angular momentum, this doesn't happen. It's possible to measure angular momentum without contributing any angular momentum to the system being measured. And it is possible for two particles to be created under conditions that cause them to be heading off in two opposite directions, with equal and opposite angular momentum.

When this happens, since the measurement can't contribute the missing angular momentum to account for a discrepancy, it seemed, from the formulas for quantum mechanics, that conservation of angular momentum and quantum mechanics would have to take precedence over relativity: that, when measured at a large distance, two such particles would have to always measure as having exactly equal and opposite spins.

This was first noted in a paper by Einstein, Podolsky, and Rosen. It was advanced by them as a paradox; a result indicated by quantum mechanics that could not possibly be true, which therefore indicated that something was wrong with quantum mechanics.

If each of the two particles carried with it a set of instructions for what to do under every possible condition of measurement, then angular momentum could be conserved without apparent faster-than-light communications (or "nonlocality"; the faster-than-light communications are internal to Nature, and not directly exploitable, so there are those who object to the use of terms implying we know something to exist that we cannot directly touch). This is known as a "hidden variables theory". Bell's Inequality notes that if that were the case, but the spin on both sides was measured not for being up and down relative to the same direction, but relative to a slightly tilted axis on one side, the chance of the spins being opposite would be proportional to the angle of tilt. But given the way a particle with a spin normally responds to measurement, the chance of the spins being opposite should be much smaller, and proportional to the square of the angle.

In any event, the experiment was finally actually performed, and J. B. S. Haldane recieved partial vindication: the Universe was proved to be strange enough to cause us trouble in imagining it. A pair of particles having opposing angular momenta, as a result of the method of their creation, whose angular momenta were in no way observed on the way to the separated detectors, thus leaving them in a pristine quantum state of entanglement, really did behave as quantum mechanics predicted: always having the same angular momentum when checked for angular momentum in the same direction, and with the chance of a difference being proportional to the square of the angle of tilt between the two detectors otherwise.

This can be used for forming a one time pad made up of ones and zeroes as follows:

A suitable radioactive source is set up halfway between two correspondents.

Each chooses, randomly, to begin measuring vertical or horizontal angular momentum.

After a particle is detected by one party, that party then randomly chooses again which type of angular momentum to measure.

Then, the two parties can communicate, over an open channel, the times at which they detected particles, and what kind of angular momentum they were measuring for at the time of the event detected.

Finally, ignoring all events except those detected at the same time by both parties, and additionally only counting those where both parties were measuring for the same type of spin polarization, a one-time-pad can be generated by taking Up to be 1 and Down to be 0, and Left to be 0 and Right to be 1, for example.

If an attacker sends particles with a known spin out, then if it happens both sides are measuring for horizontal polarization when he sent the particles out with a fixed vertical polarization, for example, the two parties may have mismatched digits in their one-time-pads, thus encountering a garble and therefore detecting tampering. An attempt to find the polarization of one of the particles, and replace it with one of the same polarization, will also fail if the wrong direction is chosen in the same way.

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