A protocol of the form you have described does exist and is known as superdense coding. It allows Alice to transmit two classical bits to Bob by sending only one qubit through a quantum channel, providing that they already share an entangled state. However, this requires that they already share an entangled state. In other words, at some point in the preparation, one of them would need to create the entangled state and send half of it to the other using a quantum channel. After we take this into account we see that using this protocol we can send 2 classical bits with two uses of a quantum channel, but one of these uses can be done beforehand.
Can we do better than this (ie send more classical bits to Bob)? A result known as Holevo's bound says no. This result tells us that the maximum amount of classical information that can be extracted from a qubit is one classical bit. This is where the Bob's 'extra' bit of information came from.
How does this relate to the Shannon-Hartley theorem? I'm not too familiar with this theorem, but looking at wikipedia it seems to only deal with communication through classical channels. Naively, superdense coding would allow you to send 2 bits of information with one use of a quantum channel, 'breaking' the Shannon-Hartley theorem. However, as we have seen, this relies on using a quantum channel to exchange quantum information between Alice and Bob. Since Shannon-Hartley does not deal with quantum channels, it doesn't really apply in your situation.
Hope this helps!