Superdense encoding allows us to transmit the information of two classical bits using a single qubit with a pre-shared Bell state qubit pair. We can duplicate the construct to transmit $2n$ classical bits using $n$ qubits. My question is: can we do better by using fewer qubits by leveraging multiqubit entanglement?
Superdense coding applies $I, X, Z, XZ$ operator on a single qubit to get $4$ different outcomes. For $n$ qubits, if we can only apply these four operators on individual qubits, then $4^n$ combinations are possible. Given $4^n=2^{2n}$, one can only get information of $2n$ classical bits. I know it's a big "if". Is that assumption correct since Pauli group forms a basis for all operators?
Sorry I am a beginner in this subject and am not clear about many concepts yet. Also apologize if the answer is already embedded in some earlier thread. Thanks.