Hierarchical relationship of quantum teleportation protocols

Entangled states are the resources of quantum teleportation. In this thesis first, the teleportation protocols of two, three and four level systems are studied. Then, the identification of their relations by teleporting a low level system with high level system protocol is examined. We explained protocols basically with two different methods. One of the methods is to perform projective measurement in the basis of generalized Bell states of particular level, and the other method is to operate a gate array. In both methods, we call the states which can be obtained after a measurement, as recovery needed states. These are the states before recovery operation. In the teleportation of an arbitrary qubit, there are 4 recovery needed states and recovery gates. In the teleportation of an arbitrary qutrit, there are 9 recovery needed states and recovery gates. To identify the relation between the protocols of three-level and two-level system, we teleported an effective qubit by using qutrit teleportation protocol. As a result, we saw that it has 9 recovery needed states and has nothing in common with the qubit protocol. Thus, there is no hierarchical relation in between. Teleportation protocol of the ququart of two-qubit has 16 recovery needed states and recovery gates. To identify the relation between four-level and three-level, we teleported an effective qutrit by using ququart teleportation protocol. We saw that it has 16 recovery needed states and recovery gates and there is nothing in common with the real qutrit protocol. Thus, there is no hierarchical relation in between. When we teleport an effective qubit by using ququart teleportation protocol, it has 8 different recovery needed states and recovery gates, and 4 of 8 is isomorphic to the qubit teleportation protocol. Ququart protocol is not reduced to qubit protocol but involve qubit protocol as a subgroup. Generally, hierarchical relationship may not be seen in the transition between relatively prime levels. However, to reach a general result about hierarchical relation, higher-level transitions should be analyzed, in my opinion. In addition to identify the relations, we also studied on effective teleportation. To teleport effectively, living in the subspace must be guaranteed. State to be teleported must live in the subspace where it belongs to, throughout the teleportation. Finally, it is important to write a specific protocol to each case for an efficient transmission.