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Suppose Alice and Bob share $n$ copies of a noiseless quantum channel $I_{A\rightarrow B}$ which can be used to send quantum states and $H_A\cong H_B$ i.e. the input and output Hilbert spaces are the same dimension, $d$. For simplicity, we can assume that $d=2$ so we are dealing with transmitting qubits.

Suppose there exists a protocol that uses $I^{\otimes n}_{A\rightarrow B}$ (and possibly other no-signaling resources such as shared entanglement) to simulate $I_{A\rightarrow B}^{\otimes m}$, where $m>n$ but this simulation occurs with some error $\varepsilon\in (0,1)$.

What theorem in quantum information, if any, bounds achievable $\varepsilon$? It seems natural that the case of $\varepsilon = 0$ is definitely out of the question since otherwise, we could simply repeat the process and communicate arbitrary amounts of information starting with just a few perfect channels. I am not sure about the case where we allow $\varepsilon\in (0,1)$.

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Not an exact answer to your question, but a possible way to get some upper bounds. Look into MacKay2003 (~page 29). He uses capacity of classical channels to get upper bounds for quantum channels. For classical channels, "capacity with allowable error rate" is known; so you can trace the process in the paper to get similar results for what you're after. My guess is that will probably take some effort though. Hopefully someone else knows where that's been worked out already.

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