Let say Alice wants to pass her qubit to Bob. By no cloning theorem, we know that Alice cannot duplicate her qubit to Bob. However, she can “teleport” her qubit to Bob if they share a Bell state .
First, Alice will apply CNOT gate from to her qubit in the shared Bell state. This will result in a combined state of
where three bits are respectively the state she wants to teleport, her qubit in the Bell state, and Bob’s qubit in the Bell state. Now, if she measures her qubit in the Bell state with the standard basis ( and ), the joint state will become
if the measured qubit is
if the measured qubit is .
Now, Alice may call Bob on the phone and tell him what she has measured. If it is the latter case, Bob can apply the bit flip gate to his qubit and result in the state in the first case. So after this operation, they will always share the quantum state
Now if Alice applies Hadamard gate to her remaining qubit, it results in
and if she measures it with the standard basis, the remaining state (i.e., Bob’s qubit) becomes
if her measurement is and
if her measurement is .
At this time, she can call Bob again and tell him the result, if it is the former case, Bob gets the original state precisely. For the latter case, he can further apply the phase shift gate to his qubit to recover .