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Consider a classical channel $N_{X\rightarrow Y}$ which takes every input alphabet $x\in X$ to output alphabet $y\in Y$ with probability $P(y|x)_{Y|X}$. It is stated in many papers that even if the sender and receiver share entangled quantum states (or even no-signalling resources like a Popescu-Rohrlich box), they cannot increase the asymptotic capacity.

What is the proof of this statement?

Notes

A half-answer to this question is in this work, where the authors comment

By itself, prior entanglement between sender and receiver confers no ability to transmit classical information, nor can it increase the capacity of a classical channel above what it would have been without the entanglement. This follows from the fact that local manipulation of one of two entangled subsystems cannot influence the expectation of any local observable of the other subsystem

However, this is a little mysterious to me since it is known (see here) that in other regimes (one shot, zero error, etc.), entanglement between the sender and receiver can be used to boost the capacity of a classical channel. However, all these advantages die away in the case where one uses the channel $n$ times and $n\rightarrow\infty$. Why is this so?

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