Proof that Entanglement Cannot Increase the Capacity of a Noiseless Classical Channel
It follows directly from our formula for entanglement-assisted capacity. An alternate proof uses the classical reverse Shannon theorem and the fact that entanglement does not increase the capacity of a noiseless classical channel. This last fact can be proved by showing that if it did, you could use entanglement to transmit information with no communication. I don't remember whether the alternate proof is in the linked reference or not.