[MUSIC] Let's summarize what you have done so far. We started from quantum theory and quantum theory contains state dynamics measurements. Quantum theory has been presented as a frame of information processing. Quantum state means preparation and quantum dynamics means a quanta process and quantum measurement means readout of qubit. Then single qubit is introduced and two qubit has been introduced. Two qubit means is a two quantum systems containing certain correlations and the correlation is called entanglement, which do not have any classical counterpart. Then we apply single and two qubits for computation and communication. We have seen that quantum computation is more efficient than classical computation, and quantum communication has some advantages over classical communication. In quantum computing, the key element is the quantum circuit. Quantum circuit can realize any quantum state transformation. Within that, the key part is a CNOT gate. I want to tell CNOT gate because any quantum circuit can be efficiently approximated by just a sequence of a few number of circuits and one of them is a CNOT and the rest is a single qubit operations. As a CNOT is introduce the interaction between qubits and single qubit notation means they change of our basis. If you have just a single qubit operations, we cannot generate entangled state. In quantum computing, I want to emphasize that the key element is in fact this CNOT gate. In experiments, if we can realize the CNOT gate with high precision, and also we can concatenate the CNOT with a very high precision then we can realize quantum computing in practice. This is in fact the main challenge in quantum computing. Then in quantum communication or in quantum channel. We have identified quantum channel in terms of bipartite quantum state. This is thanks to the Toyama Kowski isomorphism that tells us channel stated duality. If we have a quantum channel, it can be related to bipartite quantum state and vice versa. Then this map I wrote identity, which is the mapping that piece of Rho to Rho for any given unknown state Rho. This is not just identity is a preserving sequencing. There cannot be transformation identity that maps identity to Rho times of Rho. This is not just identity, but also preserving the privacy or secrecy. According to the state channel duality, the counterpart of this map is this guy, which is this state, which is entangled state. Then recorded this guy in facts is equivalent to a CNOT gate because to generate this entangled state, we need a CNOT gate. Recall that we have a CNOT gate here and if we prepare 00, then the output state here is Phi plus. The fact that we have Phi plus here means that this CNOT gate works very well. Then there's really correspondence between a CNOT gate and an entanglement here. Entanglement is an evidence that quantum circuit works well and also quantumly. The circuit this one, means that entangled state can be generated. There's a very close connection between entanglement and the power of quantum communication and quantum computing. More than that, entangled states are in fact very useful to solve some difficult problem in practice so one of them is a distance limitation. In practice, in quantum communication protocol, we use photons because photon hardly interact with the environment. Once they generated, so say for instance suppose that two photons are generate and prepared in state phi plus and the goal is to share this guy, and therefore this is sent to the other party we call b. Then this is a natural and physical limitation. Photon may disappear. There is some probability that photon appears in this side or it disappears. Then the probability that the photon appears on this side, in fact, decreased dramatically and it depends on two parameters. The first parameter, Alpha, is from the environmental condition, for instance, the optical fiber or some environmental noise, and L is the length, the distance. As distance increase and the probability that the photon center from here, arrive on this side decreased dramatically. A practical limitation is the L is about 100 kilometers. Well, 100 kilometers is not a big number. Then long-distance quantum communication is limited, in fact. One way of solving this problem is, maybe we can think about signal amplification. Signal has been prepared and then maybe it lose energy, then it becomes narrower or lower, then what we do is to amplify the signal and resend it. This process in fact is not possible in the quantum world because this amplification means a longer signal and generate a larger one. This corresponds to quantum cloning, and it's clear that quantum cloning is impossible in quantum information processing. Then we need some different way. This can be solved by sharing this entire state. How does it work? Imagine just two photons, they are disputed already as supposed in this five plus and maybe the distance is not that big, so some short distance here, and then there's another photon. This is shared with this guy, and they are also in this state. If we think about this distance and then sharing two copies, we have twice longer a distance. Then what we do is to perform a teleportation. Let me label here A, B, R, and B. What R does with the B is to teleport this guy to here by sacrificing the entangled state of this guy. Then what happen is that this guy appears here and the distance over which entangled state is shared is extended twice, so then A and B. Then the B can go further by adding more number of entangled states and etc. They can share entangled state over in any distance, and all this is possible by sharing this guy and performing a teleportation in each node. This step of teleporting this guy to here, means entanglement swapping. In the end, and this guy is entangled to this one. This come here. Entanglement is a key resource that make it possible to get certain advantages in quantum communication and quantum computation. Maybe the CNOT generate entangled state in quantum algorithms. Beyond then, entanglement make it possible to overcome some limitations in distance in quantum communication. Then we can identify entanglement is a resource in quantum information processing in general. In general means for some known quantum information task or even some unknown quantum information task. If we have entangled state, we know that entangled states are useful and that will be used for [inaudible] We now identify entanglement as a resource. Then the next question is, how can we find entangled state and how can we quantify entanglement? That's the main topic here, the entanglement theory. The main question is, given quantum state, and how entangled it is, and how useful it is in information processing. That was the main topic and the question in this lecture.