Information processing the quantum systems, it can be caught and computing or as an information task. It can be seen as a transformation of initial state to output state, and always the global dynamics is described by a unitary transformation. Now we want to look at subsystem. This is the system that we are interested in. We can write down here is a system and the rest which we are not interesting, they can be collected as environment. Your system and environment. Now we are interested in the dynamics of a subsystem here. Dynamics of the subsystem is described by a quantum channel in general. This is simply a mapping from state to space here to some other state space here. But these must be mapping or this quantum channel must be reduction of unitary transformation. This reduction is called quantum channel and I'd like to show you how to take tries quantum channel. There could be alternative and equivalent description of quantum channel. Let me start with operator representation, or class representation. We can straightforwardly write down the description over there. We have an interaction between system and environment. This is a transformation of a system and environment. We prepare our system in states role. We also assume that our environment is prepared or is initialized in a very well stabilized state. It is well prepared or equilibrated, a single state and we just put here zero. This is the state of environment. Then it's global dynamics. But we don't really care about the environment. If you perform measurement on environment, it can be outcome, but we don't care about the condition. We just average over the environment and then we can find the resulting state in the system. We perform the partial trace on the environment. This describes the dynamics of the system of interest. The next step is to simplify this expression following the definition of the trace. We perform the measurement in the environment. Then we can write down this part as rhos and zero, zero, [inaudible]. This u integral transformation is over system and environment and this environmental status act on the space of environment. Therefore, this just can be simply written as part. This part is not a number, it is some operator because this unitary transformation works on system and environment, and this inner product happens in the environment state only. We just write down here as this part as K_I. Then we write down here the dynamics of system, K_I role and K_I and here deca, we have this one. Then this collection of operator KI. This is an operator from the Systems page to system and, this collection is called a cross operators. This describes the dynamics of subsystem only. If you have a quantum channel, a quantum channel can be described by a cross operator and, once you know the cross operator, you know how to write down the quantum channel. Cross operator satisfy some specific condition and it is the following. Take two cross operators and then let's check how this looks like. We have a KI here is a definition, then we have E and U, deca and I, all environmental state. This summation works on this guy and sum of all identity are all the diagonal element corresponds to Identity and, then we have wrote once again, this is the identity on the environmental system. Overall, we have the identity of a system and environment, but then we have re-performed the inner product on the environmental state here, E and E and, the output is Identity of a system. This cross operators, if you add all of them, then it gives you or it must be Identity. This also means that this cross operators, I take this guy, since the sum all of them is Identity and this must be non-negative because is a multiplication of two number after you take the complex conjugate and deca or hermitian conjugate. This means that, this is cross operator and their products, this guy form a measurement or pure VM element. Class operators give you a description about the dynamics of sub system and they're multiplication, a product corresponds to actually construct a pure VM element so it is a measurement. Somehow these quantum channel give you a general way that a system interact with the environment or system. Here we have a measurement description and we have class operator description. One remark is that given quantum channel, we can always find a cross operator, but the collection of class operator is not actually unique. For given quantum channel, we can find many possible cross operators that describe the same quantum channel. The next is maybe we can look at example, we come back to this scenario. Think about the unitary transformation is a senile gait. It works like this one and X operation. Here this rocket works with this particular system and environment. Perhaps we are interested in the cubit in the first register, which is of our system and, then the next task is to find this cross operators. Kraus operators, we have an I here and 0 here. So 0, these environmental apart, environmental apart. Kraus operators, the next one we have I in this here, so we have a one. It's easy to calculate and apply this 0 to this guy. Then first this x operations flip 0 to 1 and then we have 0 overlap. This one but remained the same. Then this conclude the k_0 corresponds to this kraus operator. K_1 zero applies to here, but then it comes to orthogonize. We have a 0 over lab, but this one act on this X operation and it flipped it, and therefore we have non-zero overlap. This is to remain a kraus operator here once. It is clear if you compute k_0 plus k_1, k_1 is identity of our system space. This fulfilled a condition here, and each of them is non-negative. Then if we perform the [inaudible] here, then the evolution of quantum state in the subsystem can be described by wall. This is a state on this system part. This should be one wall. Therefore this will take these diagonal elements, Rho 00. That's what happened in the system pattern. We can also describe quantum channel in different way or equivalent way. It is called isometry. Here we have U and this 0 [inaudible] is environmental part only. We can define this guy as a V. This huge transformation is on system and environmental forces. We have this part. This is a mapping from system to system and environment both. Then if you introduce V as this guy, that we can rewrite the quantum channel as praise E and V Rho, and V. It's clear that V is a mapping from system and system to environment. If we compute S to SE and performing this guy and S to SE. We have the definition here. This will be identity of a system as well. You can get previous as if you apply vector here, then a complex system to environment and system 2, environment to back to a system. Overall it works like a system identity on the system part. This free, this guy is called isometry. Quantum channel can be described by kraus operators and equivalently with isometry. Depending on the context, sometimes is useful to apply isometry representation, or sometimes is useful to apply y kraus operator presentation.