[MUSIC] Quantum channels are special. So quantum channels are different from classical channels. So what the main difference is, is the monogamy property. So we call that the quantum channel can be monogamous, it means the following. So this follows from its correspondence by part of the quantum channel, so by part of the quantum states. So this belonged to, This is space. And if this is positive and this is complete positive and this is quantum channel, then this must be a quantum state. So we have quantitative correspondence between these two guys but their correspondence also holds true for qualitative things. So here we had say, this quantum state, say A over AB, and we call it, I mean, the map that corresponds to this quantum state was identity channel. So we call that we had A and B and C, and the states here corresponds to this mapping. So take this state, and we can ask, if we label here B and C, can this state B also entangled to some other states? As a simple example we can imagine these states, you can call this is actually Ghz state 00 plus 11. So clearly this is a tripartite entangled state, and they are fully connected. But whenever you look at this guy by tracing out the third one or one of them, this guy is 1/2 00 and 11. So this means the state can be prepared by LCC and it doesn't contain entanglement. So it's a separately state actually. So to create the entanglement of three pattern system, also this way, then this state must be separable, entangle cannot be seen in this bipartisan system. Now if you look at this guy, can you find or say is it possible to find the tripartite state? So we can include E, such the overall state, BCE such that whenever we trace out E for this part, this comes to 5 plus. I mean, what kind of state will be like here? So the answer is very simple, I mean the answer is actually the only possibility is this guy for some state E. So between this partition BC versus E, we have a tenth product. So low entanglement is possible between BC and E. Again, this means that if BC is entangled in this way, then B cannot be entangled any other parties, C as well. So if C is entangled with the B, then C cannot be entangled with any other parties. This property is called monogamy. So one party can be entangled or can share this state with only one party, no more than that. So this property is clearly monogamy. And now we want translate this one to quantum channel. So the property of quantum state like a non-share ability or the monogamy property and how this can be translated to this one. This one corresponds to this guy, identity map. So if we see the identity map, this is a mapping from row on any side, say N row. So this will keep the single state to a single state. Maybe one can ask is it possible to dispute or can we imagine this kind of thing A2 say C and C prime such that row is map into row and row. So like a copying. So one state is mapped to both sides C and C prime is equal to copy. This is precisely identity mapping because it preserves the same status here, but it generates two copies because of 10. And we know that this is impossible because quantum cloning is not possible and this is forbidden by the no cloning. So the monogamy property of quantum state here immediately implies that quantum cloning is not possible because the channel itself is like monogamous properties. And is concerned with some secrecy property. If you prepare quantum state, this channel is responsible for sending the state in a secret manner, in the sense that nobody has the same copy. I mean, the fact that nobody can have the same copy, yeah, this follows from the rows of quantum mechanics. So quantum channel. So instead of classical channel, if we apply quantum channel, it is something special. At least it can preserve some secrecy property. And this is closely related to the no-cloning theorem. Then what will be the questions about quantum channel? So quantum channel in the end it's just kind of transformation, so it's a mapping from some certain information entities on A to B. So what are the information entities here? Say first bits. So I write here bits. Bits means classical information or information can be described by bits classical information. So also the information can be described by classical systems. And the next one is a cubits. This is quantum bits and quantum information and some certain information to describe that cubit is necessary. Then we need a cubit. Also qualitatively differently, I mean, we have secret bits. So secret bit is classical bit, but is bits nobody knows except the legitimate parties of A and B. So these are different kind of information. Then in the other question about this channel is that how useful the channel is to deliver or reliably transmit this information to B side. So the quantification often we call channel capacity. So we define the channel cabinet like this and then this channel is repetitively used or IID manner, then one has to think about this repetition. And then we are interested in certain quantity, I write here Q, sorry, Q. So we are interested in this Q quantity and Q depends on which information we want to send, the bit or cubit or S bit. And yeah, we want to regularize this quantity of N, and so this will define channel capacity. So the first one if we want to send bit only, that we call classical capacity. A second one if we want to send a cubit, so how many cubic can be reliably transmitted by using this channel? Then what is the maximum weight at which the cubit can be efficiently and reliably sent from A to B? And this define a classical quantum capacity. Or quantum channel capacity. And third one, if we want to send a secret bit and this is concerned with the private capacity. So we are plugging these different quantities here then we have a quantum channel capacities. And this classical capacity is concerned with the whole level quantity. So which means we prepare cubits for the proposal of delivering classical messages. So we include our classical message to quantum states, and we optimize our measurement to extract that classical information, and that is concerned with the whole level quantity. And the quantum capacity is closely related to this property. So quantum channel cannot dispute quantum state to arbitrary many parties. And, Yeah, this is closely related to this one, and the channel capacity is often called coherent information. And this private capacity is concerned with the cryptographic properties. So how many secret bit can be reliably sent to and then this is quantified by private capacity? And in a more general picture, I mean, this is from single center and single receiver. In the most general picture, I mean, this quantum channel can be generalized for instance look at just two parties centers, so two centers and there will be two receivers. So yeah, AB and CD, so there are now four parties and here also we can apply this description about the quantum channel. And then there are many, many different measures that the first one is that A's want to send information from A to C, and B is to D. So then we want to maximize the information from 8 to C plus B to D. And this is called interference channel. Or A and B they want collaborate each other and they cooperate and then they want dispute messages to CD both, and this is called broadcasting channel. Or AB and they are located far apart and then they send message to BC together. And this is called multiple access channel. And yeah, this define the network channel and more generally there's a multiple input and multiple output is called memo or memo channels. So in general we have a network channel and for that there are lots of lots of questions we ask about how useful quantum channel are. And then radical difference of quantum channel is compared to classical counterpart. In this network part, it has been shown that quantum channel capacity, say the quantum network channel capacity, is higher than in the classical one is important, and quantum channel capacity can be activated. So this is something we don't have in classical case, but we do have in quantum case, we do have activations phenomena. So I want to tell that, [INAUDIBLE] so the quantum advantages in communication is activation. For instance channel capacity can be activated, And the channel capacity is higher or can be higher. Higher channel capacity and third one secrecy. Yeah, so these sequences somehow generate in the sense that the channel itself is monogamous. So this is generally, and then quantum channel is dealing with entangled states. So it introduced some complicated correlations, we do not have in classical void, therefore it turns out that it is possible to have a higher channel capacity. And interestingly, I mean, this activations as possible and this also thanks to entanglement, sometimes activation means we have channel capacity, take two channels like this, and this can be certainly higher than each of them. So each of them maybe even 0, but combining together we have even larger channel capacity and this is called an activation. And in the network channel we have a higher channel capacity and quantum channel genetically preserve some properties of secrecy.