Quantum teleportation can be seen in two pictures, and let's look at this one first. We have constructed this picture by translating one-time pair into quantum analogy. We have assumed that two parties, Alice, and Bob, they share at least five process states. This is equal superposition of 0, 0 and 1, 1. Then we put the messages state here m. Then the first step in quantum teleportation is that Alice perform CNOT operation on these two qubits, and then Alice apply the Hadamard transformation in the qubit, in the first register for the message of qubit, and then perform measurement. After this quantum operation, and second measurement, Alice has to collect the measurement outcome X_1, X_2, and simply put X. This is two-bit information. Then the measurement outcome should be reported to Bob's side, and Bob will choose this local data transformation. This is still unmeasured, and according to the measurement outcome, Bob will choose the local inter-transformations. Here, identity X, Y, and Z these are the Pauli matrices. By doing that, this messages state m will be completely recovered on Bob side. This quantum teleportation, here, the message m is transmitted from the location A-B. Without really sending the qubit from ones to the other, but just by sacrificing this maximum interstate 5 plus. That's the message of quantum teleportation. In the first type, it is maybe surprising, but this is actually completely analogy of the one-time paired. In one-time pairs we put here a secret bit, and the message here is not merely send it to Bob, but by sacrificing the secret bit here, the message has been sent to Bob in a secret manner. These can be seen in an equal picture like here. We can say that this is space, and this is time. We can think of two qubit here. They've generated by interaction or CNOT gate between two qubits in the beginning. These two qubit state are prepared. This state is disputed over some distance so this qubit equals to A Pi, and this equals to B Pi so we have A here, and B here, and this is the messages part. These two qubits, both located on A site. The first step is to apply the CNOT gate, and then a Hadamard transformation, and then local measurements, this measurement in the computer database. We will get outcome 0 or 1, and 0 or 1, and we collect the measurement outcome X. Again, X_1, and X_2, and this outcome should be reported to both sides. This transmission can be done by classical communication. According to the measurement outcome on A side, Bob can choose one of the four possibilities in order to recover the message state here. That's a quantum teleportation. These parts, in fact, perhaps you can think of collective measurement or jointed measurements, so we perform measurement collectively. This measurement actually corresponds to Bell measurement, so measurement in the bell basis. You can easily see then how these individual measurement corresponds to Bell measurement. Sometimes or actually in the scenario when quantum teleportation has been introduced, here. Alice, what she does is actually bell-measurement. The bell-measurement, and the outcome would be Phi plus or Phi minus, or Psi plus or Psi minus. Up performing the bell-measurement on a site and Alice will have one of the four outcomes, and the first outcome is the detector described by these basis shows a click event, and second, third, and so on, etc. Then at least we report the measurement outcome is one of the four possibilities. According to measurement outcome, Bob applies local operation. Then you can find the correspondence. If Alice get measurement outcome Phi plus, then Bob doesn't apply anything, just remaining identity, then that is already the message state. If the outcome is Phi plus, then Bob will apply X operation to transform the Bob state into the message state, and Phi minus then Bob will apply the local operation Z, and Psi minus then is a poly-matrix Y. In this way, Bob can recover the message state. We can easily check by calculation. Message states, and the entire state here, so M and AB. Let's write down the message state for n Alpha and Beta, like this. This Phi plus is this guy. You can expand this one. Then the first two QBs belong to Alice side, and the third QBs belongs to Bob side. We can just rewrite this one as 1/2 now. Alpha, and this guy is simply Phi plus and Phi minus and zero and Beta. This guy is Psi plus, Psi minus zero. The third one, Alpha plus and minus, and we have a state 1 on p side. Then you can collect all of them 0.5 or Phi plus, and then Phi plus, we have here and here, then Alpha 0 plus 1 and 0.5 Phi minus. Then here, we have Alpha 0 minus 1 plus 0.5 Psi plus. This comes from this guy. Beta 0 plus Alpha 1 and 1/2 Psi minus and 0 minus Alpha. If Alice performed bell-measurement, then the outcomes appears one of them, and the probabilities is one of four, so equal probability and with probability 1/4, Bob, this will either state one of them; this one, this one, or this one, and this one. Depending on the measurement outcome from Alice side, Bob knows which local operation he has to perform. According to this table, then, by local linear transformation, Bob state would be the message here. You can check out how these schemes can teleport the message state on A side to B side.
