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The experiment takes advantage of one of several phenomena that describes quantum mechanics: entanglement, or "spooky action at a distance," as Albert Einstein called it. When two particles are entangled, they remain connected so that an action performed on one affects the other as well, no matter how far apart the two are. In the same vein, when one measures the state of one particle in the entangled duo, you'd automatically know the state of the second. Physicists call the states "correlated," because if one particle — a photon, for example — is in an "up" state, its entangled partner will be in a "down" state — a kind of mirror image. (Strictly speaking, there are four possible combinations for the two particles to be in).
The weird part is that once the state of the first particle is measured, the second one somehow "knows" what state it should be in. The information seems to travel instantaneously, without a speed-of-light limit. [8 Ways You Can See Einstein's Theory of Relativity in Real Life]