# In standard quantum teleportation, when can quantum states be distinguished with certainty?

Through a few examples, I'd like to learn under what circumstances can different quantum states be distinguished from each other.

So for example, the standard quantum teleportation scheme starts out with the three qubit composite $$(\alpha\vert0\rangle + \beta\vert1\rangle)(\vert00\rangle+\vert11\rangle)$$ and arrives at the state $$\sum_{a,b=0,1} \vert00\rangle \otimes (\alpha\vert a\rangle \pm \beta\vert b\rangle)$$. Then a measurement in the computation basis determines the state of the subsystem comprising of the first two qubits, the outcome of which determines what quantum gate is applied to the third qubit, moving the third qubit into the target state $$\alpha\vert 0 \rangle + \beta\vert 1\rangle$$.

What is the measurement that is performed on the three qubit system that determines the state first two qubits? More generally, when is it possible to determine the state with certainty?

• I'm a bit confused by the notation. In the standard teleportation scheme the state over the first two qubit will be one of the four Bell states, see e.g. the wiki page, so why do you write it as just $\lvert00\rangle$? – glS Apr 25 at 12:54