Can superdense coding be made more efficient?

Question

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.

Answer 1

No, you can't send more than 2 bits per transmitted qubit. Ultradense Coding would allow FTL Signalling.

The basic problem is that, if either teleportation or superdense coding was slightly more efficient, iteratively nesting them inside of each other would allow you to send more encoded qubits than the number of physical qubits you sent. Suppose you could ultradense code 3 bits into 1 sent qubit. Send 2 ultradense qubits, equivalent to 6 bits, but use those 6 bits to power teleportation of 3 qubits, which are actually ultradense coding 9 bits, which are teleporting 4 qubits, which are coding 12 bits, which are teleporting 6 qubits, which are coding 18 bits, teleporting 9 qubits, coding 27 bits, teleporting 13 qubits, coding 39 bits, ... you get the idea.

Throw an error detecting code on top, and the receiver can just guess at the initial teleportation bits and see if it worked. This would presumably allow them to receive the message before you even send it, which is clearly a problem.