# Tradeoff between error and rates of quantum communication

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 $$\varepsilon\in (0,1)$$.