quantum mechanics - Extending the idea of superdense coding

I was reading through the superdense coding protocol, that lets A convey two classical bits to B by sending one qubit (assuming B sends A a qubit beforehand). So B creates a 2-qubit state and sends the first qubit to A. A performs a transformation on this qubit and sends it back. Based on that, B can distinguish whether the two bits of A were 00, 01, 10 or 11.

The question is, why can this idea not be extended to convey more than 2 bits? Can A convey n bits to B this way? More specifically, can B create an n-qubit state and send the first qubit A? A can then apply a unitary transformation on this qubit and send it back to B, who can distinguish between the 2^n possibilities and figure out which n bits A had.