michael_nielsen

The interesting thing for us is that the message d+r from Alice to Bob is completely uncorrelated with the data d. In other words, that message tells us nothing whatsoever about the identity of Alice's data. (This is also why it's cryptographically useful, of course!)

This seems promising: Alice and Bob now share an entangled quantum state with the amplitudes a and b. Can they do something so Bob ends up with the original state a|0>+b|1>?

Unfortunately, Alice can't do it by simply applying a quantum gate - the state will remain entangled. But maybe she can make a measurement of some sort?

Suppose Alice measures her remaining qubit in some basis |m>. The resulting conditional state for Bob is, up to normalization: a<m|0> |0> + b <m|1> |1>

Ahah! This is very promising! If we simply choose the basis |m> to be the equal superposition states |+> = (|0>+|1>)/sqrt 2 and |-> = (|0>-|1>)/sqrt 2 then Bob will get the following conditional states: a|0>+b|1> a|0>-b|1>