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The classical part means sending the result of a measurement, so 0 or 1 and not some state like $\alpha|0>+\beta|1>$ (where the qubit is a 0 with probability $\alpha$ and a 1 with probability $\beta$ - even if it's something that closely resembles classical information like $1|0> + 0|1>$, so a 100% probability to measure 0 - but that's still ...


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If you teleport a qubit in a known state you can test whether the qubit at the end of the teleportation produces the right probabilities for measurement of some complete set of observables. There are many schemes for doing tomography on qubits, such as http://arxiv.org/abs/1310.4492. Also, quantum teleportation is not spooky, see ...



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