Consider a fixed qubit channel $\mathcal{N}$ and some QECC that prepares a logical state $|\psi_L\rangle$ of $k$ logical qubits using $n$ physical qubits. After transmitting data through the channels, the probability of correctly decoding (and thus recovering $|\psi_L\rangle$ at the output of the decoder) is $p_{err}$ - assume operations are implemented perfectly.
Assume that I have some kind of scalable QECC scheme, so that I can express $p_{err}$ as a funtion of $n$ and $k$.
My question is, what does the tuple $(n, \frac{k}{n}, p_{err})$ tell us about any sort of capacity of the channel $\mathcal{N}$?
My thoughts so far:
- This setup implies $\frac{1}{2} \lVert \psi_L - \mathcal{D} \circ \mathcal{N}^{\otimes n} \circ \mathcal{E}(\psi_L)\rVert_1 \leq p_{err}$, so the tuple does describe a code with rate $Q=k/n$ and error $\epsilon=p_{err}$ (using definitions/notation from Wilde's textbook)
- But this doesn't seem to actually tell us about the quantum capacity of the channel, since we have no way of taking the error to zero in general
- The closest I got was using some ideas from (Tomamichel et al, 2015), that $(n, k/n, p_{err})$ gives a lower bound for the boundary of an achievable region of rates $\hat{R}_\mathcal{N}(n, p_{err}) := \max \{R: (n, R, p_{err})\text{ is achievable}\}$ (where "achievable" just means that tuple describes a code).
I would like to know if (3) is the strongest statement we can make about $\mathcal{N}$ in this scenario, or if there are other capacities that I missed that are bounded once we achieve this tuple.
