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My understanding is that current quantum teleportation methods need a classical channel to transmit information (and so is limited to the speed of light). See the Protocol section of Wikipedia - https://en.wikipedia.org/wiki/Quantum_teleportation . To my knowledge, there is no particular reason to believe that this limitation can be removed.

The simplest way to see this is to consider a state such as a|0> + b|1>, which is the quantum state of the particle we wish to teleport from Alice to Bob. If Alice does the measurements needed to teleport it, Bob might end up with either a|0> + b|1> or a|0> - b|1> (or two other similar states) with equal probability.

To actually get the correct state we began with, which is the only way to transmit something meaningful, then he needs the results of Alice's measurements to make a correction. These can only be transmitted at the speed of light, so speed of information transfer is still limited to the speed of light.

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