We want to apply our descriptions about quantum channel to qubit as a specific example. Qubit in general, can be characterized by this vector. We are interested in how this, say, if you start from a point in a Bloch sphere and now it is transformed within the Bloch sphere. The first case we are interested in is bit-flip. We can consider channel and leave the state in the same with probability one minus Lambda and roll, and we'll get a probability lambda. It comes to some error which is generated by this x, bit-flip error, so we have this guy. This channel is called a bit-flip channel. We can apply the different descriptions to this guy. This can be seen as cross operators, so I can write k_0 Roh k_0, and k1, Rho k_1. We can check that some of them, say k_0, k_0, k_1 is identity. We can describe this channel as cross-operators. It can also be described by the teleportation picture. Here we have this one and this one. Your P is the projection onto the Phi plus state. What is this guy? We can find that this 23 is Identity, and this channel here, and this one. Applying this channel to this Phi plus state and what happened is with probability 1 minus Lambda, it remain the same state, this guy. With probability Lambda, we have bit-flip, this one, which is Psi and plus n plus. This state is the state that corresponds to this quantum channel. If we plug in the Psi in quantum teleportation, so this c_23 is shared, and we perform better measurement and we apply state rolling this guy. State we have after all performing measurement and this guy, and the state we have here. Bit-flip channel can be seen in different formulations. Let's see how this is evolved by this channel. Because this is mapped to this channel, what we are interested in is, in fact, we should have Identity and Sigma, and this will be a new block vector, and we're interested in how this looks like. Suppose we start with this guy A, B, C, this would be some vector in a Bloch sphere. Then this guy plus Identity and one-half. But then we will see that with this probability, everything remains the same. But after the channel with probability Lambda, the state Rho will be flipped by this x operations, so something different will be this part. Then at this point, we have Identity anyway. Overall we should have a one-half and one-half and Identity. We have x and ax plus by plus cz and z. This is the part that something new due to the fact that we have applied a Jakarta channel Lambda. You can check that this guy, so I think it shukd be here, x because we have an x, x and x should make x's. Then, we have x and x, x will be the same, so a_x and b, x, y, and x, and it happened that we have a flip so minus y. Just as well, so we have a minus c. Then this guy vector, you can check that this new vector is one minus Lambda and Lambda a minus b and minus c. Then what happened is that a 1 minus 2 Lambda b and 1 minus 2 Lambda c. At this level, this can be any point in a Bloch sphere, so it can be any. But then here, what happen instead, a point will be preserved. A is the direction of x, so x, and this part, y, and this part, z. Then say all possible values in this axis is okay. But why? The range is limited because previously b can be any value between minus and one. But then now because of the channel, the minimum is 1 minus 2 Lambda and b and 1 minus 2 lambda. If it says Lambda is one-half, then b must be zero. If Lambda is small number and this is larger than minus one and this is smaller than one, which means we have a limited range in this direction and z as well. We have a limited range, say, the same magnitude, so we have here. After all the possible range of this vector happens like some ellipsoids. This is possible, this is allowed, and this is allowed, and this is allowed. We Bloch sphere, this ellipsoid is allowed after the quantum channel. For instance, after quantum channel, it's not allowed to have this state. This happens because of the bit-flip channel. There are other quantum channels for qubits. If we replace this guy as z and this guy as z, then we got a phase bit-flip channel. Probably why metrics is the case that detailed phase, they flipped it both, then it's called a bit phase flip channel. A more general form of a quantum channel is called Pauli channel. Pauli channel is the combinations of all the possibilities. This will be given though in maps to one minus Lambda Roh. Then with probability Lambda one, it comes to bit-flip. Probability Lambda two, it comes to the phase bit-flip. Probability Lambda three it comes to bit-flip errors. Lambda is sum of all of the three parameters. This channel is called Pauli channel.
