Understanding this description of teleportation

In the context of quantum teleportation, my lecturer writes the following (note that I assume the reader is familiar with the circuit):

If the measurement of the first qubit is 0 and the measurement of the second qubit is 0 , then we have the state $$\left|\phi_4\right>=c_0\left|000\right>+c_1\left|001\right>=\left|00\right>\otimes \left(c_0\left|0\right>+c_1\left|1\right>\right)=\left|00\right> \otimes \left|\psi '\right>$$.

Now to get the final teleported state we have to go through the final two gates $$Z, X$$.

My lecturer writes this as;

$$\left|\gamma\right>=Z^0X^0\left|\psi '\right>= \left|\psi'\right>= c_0\left|0\right>+c_1\left|1\right>$$

Here are my questions:

1. Why is it that the we do not have $$\left|\gamma\right>=Z^0X^0\left(\left|00\right>\otimes \left|\psi '\right>\right)$$? I don't understand why we cut the state $$\left|\phi_4\right>$$ "in half ", to use bad terminology, at this step.

2. What does the superscript 0 on the operators refer to?

3. In the final state again to use bad terminology we only use half of the state $$\left|\phi_4\right>$$ can we always assume that the final state will be the $$\left|\psi'\right>$$ part of $$\left|\phi_4\right>=\left|xy\right>\otimes\left|\psi'\right>$$ state, and if so what is the significance of the final step.

If my question is unanswerable due to a deep misunderstanding of the math processes here, I'd still really appreciate some clarification on whatever points can be answered and I can edit it to make more sense as I learn.

2. The superscripts denote the power of the operator applied: if both measurement results are 0, you don't need to apply X or Z corrections, which is the same as applying these operators raised to power 0 ($$X^0 = Z^0 = I$$).
3. After measuring the first two qubits, the state will always be represented as $$|xy\rangle \otimes |\text{some state}\rangle$$, but to convert the receiver's qubit from $$|\text{some state}\rangle$$ to the state that needed to be teleported $$|\psi'\rangle$$ you'll need to apply corrections which depend on the measurement results $$x$$ and $$y$$. I believe this is what you refer to as the final step, and its goal is fixing the amplitudes of the state for it to match the state that was teleported.
• Minor nitpicks. If you write text within a mathematical expression it's best to use the \text{} formatting; otherwise, the $spacing \space between \space letters \space is \space somewhat \space uneven$. Moreover, 1., 2. and 3. is the only format that works for numbered lists (on SE). 1), 2) and 3) does not work. Cf. help/formatting. I've edited these things; hope you don't mind. Also, it might be useful to clarify the Alice-Bob terminology in the answer itself as the OP doesn't seem to be using it. Apr 15, 2019 at 5:31