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Is it possible to get two atoms to opposite quantum states of one another so when I change the state of first one, the state of the other one changes too? Is it possible to move them to another place on Earth and communicate like this? I've seen it in a sci-fi game but am wondering if that is truly possible.

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It is a common misundertanding of the concept of quantum entanglement. In short : the answer is clearly no, it is impossible to communicate instantaneously at distance with these states, but quantum entanglement allows surprising things, like transferring the state of one atom without being able to measure it. I think (but I've not checked) that the (wrong) idea of using quantum entanglement for instantaneous communication at distance spread through Science-Fiction after the experimental verification of Bell's inequality in the 80's.

If you want more detailed answer, look at these two former answers on this site.

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