## What does it mean "the N uses of classical-quantum channel"?

1

I was reading a paper Quantum Polar codes by Mark M. Wilde, where he discusses the N uses of the channel in the classical-quantum channel setting. What does he mean by "multiple channel uses"?

In the above context I want to use a depolarization channel $$\Delta_{\lambda}$$ (https://en.wikipedia.org/wiki/Quantum_depolarizing_channel) over which $$N$$ qubits are transmitted. To accomplish this task I have two ideas given below, and I am not sure if either one of them is correct.

Two options:

1. Should I evolve each of $$N$$ qubits by $$\Delta_{\lambda}(\rho_i)$$ indivisually and then do a tensor $$\hat{\rho_i}\bigotimes\hat{\rho_{i+1}}\bigotimes,...,\bigotimes\hat{\rho_N}$$,to get final state
2. Or should I input a composite state first and the evolve it as $$\Delta_{\lambda}(\rho^{\otimes N}) = \lambda\rho^{\otimes N}+\frac{1- \lambda}{2^N}I_{(2^N X \,2^N)}$$ Isn't it like,if you send one qubit in a single use of the channel, then you send n qubits in n use of the channel? – Hasan Iqbal – 2020-12-02T01:13:59.117

1Please add a link to the reference and the surrounding context. It probably means applying the same channel $\mathcal{E}$, $N$ times, namely, the channel $\mathcal{E} \circ \mathcal{E} \circ \cdots \circ \mathcal{E}$. Or, it could mean, applying the $N$ copies of the channel on $N$ copies of the system, which would be $\mathcal{E}^{\otimes N}$. – keisuke.akira – 2020-12-02T05:33:46.717

@keisuke.akira : I have updated my post, please give some suggestions. – Najeeb Ullah – 2020-12-02T10:48:19.313

could you please show how (2) was derived? – Hasan Iqbal – 2020-12-02T18:18:28.130

Each of the $x_i$ is embedded into its respective $\rho_i$ and then Tensor them to get $\rho^{\otimes^N}$ and the rest is depolarization linear map. – Najeeb Ullah – 2020-12-03T08:09:55.463