michael_nielsen

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>

That is, Bob's state is just Z^z|psi>, where z is the outcome of a measurement in the |+>, |-> basis.