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Assume that Alice and Bob are allowed to share entanglement and are spatially separated. Alice is given an unknown state and asked to measure this in the computational basis to obtain $\vert 0\rangle$ or $\vert 1\rangle$. Is there some way for Bob to also have a copy of same state as Alice instantaneously?

Note that it does not violate no-signalling since the outcome of the measurement for Alice is random - so she cannot use it to communicate. Another perspective is that this is sort of like cloning but since the only outcomes that Alice gets are $\vert 0\rangle$ or $\vert 1\rangle$ and they are orthogonal, it isn't forbidden by the no-cloning.

If this can be done, how should she and Bob design a quantum circuit that achieves this? Otherwise, what forbids this possibility?

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Assume this works. Then, nothing prevents Alice from applying the same protocol to a quantum state that is known to her, such as $|0\rangle$ or $|1\rangle$. This way, she could send information to Bob instantaneously. Thus, it violates faster-than-light communication and thus is impossible.

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