Quantum Channel can be seen as a teleportation. Beforehand, quantum channel can be described by Kraus operators. How did we derive this Kraus operators. We first look at the computational process, the global transformation must be neutral transformation that may corresponds to some computational task. Then we were interested in the system dynamics only. We looked at the subsystem, and we trace out the environmental part. Then we have characterized the dynamics of this subsystem, and this one corresponds to quantum channel. We wrote down the quantum channel in terms of Kraus operators. Kraus operators is among the interactions over all qubit, and system, and environment. Kraus operators, is the operators that characterize the evolution of the system part only, that's the Kraus representation. Then we looked at the teleportation, this is say information processing, and this computational processing. Information processing, we interpret this information processing as a quantum channel. Initially two parties, they share some entangled state or some quantum state. They can be entangled or separable or it can be a five plus why not. Then we prepare states that we are interested here. Then we perform the measurement, and we assume that the measurement happens in this basis. This happened with some probabilities. We multiply that probability to make it normalized. Then we have our output state here, precisely the description about this quantum channel. Quantum channel can be described as a Kraus representation, and equivalently can be described in a quantum teleportation scenario. Then it seems like there's a correspondence between this guy, and this guy. Given these states, we can write down the quantum channel here. Given quantum channel here, we were able to write down this bipartite quantum state. This is not a coincidence. Actually, quantum channel can be defined in a axiomatic way, and quantum channel is a mapping from state to state so it must be a mapping from the set of quantum state into set of quantum state. This means that for all state here, it should be non-negative, and the resulting state of role, it should be non-negative. This map is a mathematical map. This map preserve the positivity. This condition is called positive. Whenever the map satisfy this relation, mapping from positive operator to positive operator they record the map is positive, it's positive map. The next one is Trace-preserving. To interpret this operator as a quantum state, we should have this condition. Then after the mapping, we should also have this condition for all quantum state. Then the trace is preserved here, and this condition is called twist trace preserving, simply we call called TP. You can compare this one to probabilities. The classical analogy of quantum theory could be stochastic process. Instead of quantum state, if we have a probability then you maps to some other vector by a stochastic process, this is called stochastic process. The conditions that we require here is that this should corresponds to some probability, and this also probabilities. This means that the elements here Pi, they should be non-negative, and sum all of them is one. The same conditions should hold to in this 11. Lambda Pi, and the element must be non-negative. If you add all of them, it should be one. Here we have a very similar structure, and this positive mapping corresponds to transformation of probability to probability. We need some more extra one, which happens in a quantum theory, not Bernardin in classical or stochastic process. It is complete positivity, we call CP. What this means, this follows from, you can see from this picture, I mean the teleportation picture. This N is described by a bipartite quantum state. I think it's better that way, C_N is an element of bipartite quantum state. Then this guy is related to this guy. This channel is described by bipartite operator or bipartite state here, and this guy characterize this guy too. There is, in fact a one to one correspondence. Once we know this guy, we can define this one, and once we know this guy, we can also introduce C_N. The correspondence between mapping from Hilbert space to Hilbert space in the bipartite system is called a Choi Jamiolkowski isomorphism or state channel duality. Here we have a state and we have a channel duality, it's called Choi Jamiolkowski isomorphism. This means there's really one to one correspondence between these two spaces. This should be a quantum state. For our channel, and this define in fact, C_N. We have a channel here and mapping from, of course this is one to one correspondence. The isomorphism from this space to here is given by this relation. The other way around, knowing this, we can construct a channel. This is precisely correspond to this teleportation picture. That one means N of some state Rho, this holds for all Rho, should be Rho 1, 2,3. Then we do project the system to this guy. Any of Phi plus is the projection in the d dimension space. Here we can write t and d. To make it clear, I have to put this one as I runs from 0 to d minus 1. The first line, this one corresponds to the mapping from to here. Second one, this guy corresponds to the other mapping here too and this is isomorphism. Then the resulting comment is that this guy is quantum state and therefore this must be non-negative. If this is positive, suppose that for this particular state of Phi plus and we apply this channel to an extended Hilbert space and is non-negative and this is the equivalent to saying that for all bipartite quantum state, this will be state on h tends h. This guy is also non-negative. Now, we can include k as well. We can consider k-dimensional extension and this happens for all k's. Then fulfilling this condition is called k positive, for all k then you have a complete positive. This is equivalent to this guy. After all, we can conclude that this map should be completely positive, CP. We have extra condition CP apart from this positivity and trace preserving. After all, the legitimate quantum channel, we can even define a quantum channel is a CPTP map, is a complete positive and trace-preserving map, defines a quantum channel. That's the conclusion from the axiomatic approach to quantum channel. Whenever we have a quantum channel, say subsystems and dynamics here, or this teleportation picture and we have a map, say a CPTP map, they are all equivalent and different ways of describing quantum channel or dynamics of subsystem.
