# The solution when we transmit a qubit through a Pauli channel?

A Pauli channel is defined as a convex combination of Pauli operators, i.e. $$\epsilon_{\text{Pauli}} (\rho)=\sum_{j} q_j\sigma_j\rho \sigma_j$$, where $$0 \leq q_j \leq 1$$ and $$\sum_j q_j=1$$. Now, I want to transmit firstly a pure state qubit through it, and then a Bell state. How do I start working on this?

Let's say $$|a\rangle$$ is the pure state qubit and $$|b\rangle$$ is the bell state.
Putting $$|a\rangle$$ through the Pauli channel depends on which of the Pauli gates (which of the $$\sigma_j$$) you want acting on it. Once you choose this (call it $$\sigma_a$$), apply $$2\times2$$ Identity operators to it on either side until it's the same size as $$\rho$$, then just do the matrix multiplication: $$q_a\sigma_a \rho \sigma_a$$, where the $$\sigma_a$$ are both implicitly including enough identity matrices to make them the same size as $$\rho$$.
The Bell state will involve two qubits, so you can choose up to two Pauli operators ($$\sigma_b$$ and $$\sigma_c$$) that will act on it. You take the result of the previous matrix multiplication and do another matrix multiplication.