This post gives a beginner their first look into quantum computing. It covers all the basics required to understand how quantum teleportation works. For the teleportation algorithm itself see the second part of this series here, and for how it is implemented using Qiskit on a real quantum computer, see the third part here. For this tutorial I will assume a basic understanding of linear algebra and braket notation. Although I will use braket notation throughout, all of this could also be though of in terms of normal matrix notation too.
As I am sure you are all familiar with, bits on a classical computer can be stores as either a 0 or a 1. What makes quantum computing different is that we can also have a bit that is in a superposition of 0 and 1.
We will define the two possible measurements as the kets |0> and |1>. The |0> state is the state in which we have 100% probability of measuring 0, and the |1> state is the state where we have 100% probability of measuring 1. All other states can then be represented as a linear combination of these.
The coefficients a and b represent the probability amplitudes of measuring each state. So, if we have this state we would have a probability equal to a² of measuring 0 and b² of measuring 1.
This basis of |1> and |0> is the most common in quantum computing. For teleportation we also need to define one other basis out of the infinitely many possibilities. The |+> state and the |-> state are defined as following in terms of the |0>, |1> basis. They are both in an equal superposition, with equal probabilities of ½ of measuring 0 and 1. The difference is solely a phase difference.
In quantum mechanics, when we take a measurement the state collapses onto one of the eigenstates of the system. Therefore for these bases there are two different types of measurement. The first is called a Z measurement and in taking this measurement we collapse the state onto one of the states in the |0> |1> (Z basis). The second is called an X measurement and this collapses the state onto one of the |+> |-> states (X basis).