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If you throw one entangled particle inside a black hole can we know what is happening with it from looking into another entangled particle outside the black hole?

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This is a version on steroids of the famous Einstein–Podolsky–Rosen Paradox. In General Relativity, to get out of a black hole requires you to move faster than light, so your question can essentially be simplified to "Can we use entanglement to communicate faster than light?" If we can, then someone inside the black hole can use the entangled pair to send information outside.

The answer is negative. Once you measure something on the particle you have with you, you won't be able to tell whether the result you got was due to something that happened to the other particle (i.e., if it is because someone tried to send you a message) or if you got the result you got by pure chance, since Quantum Mechanics is probabilistic.

Perhaps someone better versed than me in Relativistic Quantum Information will give a more complete explanation. For the time being, if you're interested in the EPR paradox, some standard Quantum Mechanics textbooks discuss it. I believe Weinberg's Lectures on Quantum Mechanics has a few words on this topic, if I recall correctly.

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