This won't happen and the paper does not speculate in any way about it.
But they are vague about the implications and I'm generally unsure about the practical relevance of quantum protocols like superdense coding ...
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Superdense coding specifically is unlikely to ever be practically relevant. Even for people quite optimistic about the future of quantum computing it is hard to imagine a scenario where a single qubit will be cheaper than two classical bits.
Other quantum protocols are much more likely to be useful, perhaps particularly for cryptography.
I sometimes consider a qubit to be worth two classical bits in a very heuristic sense. One can superdense-code two bits into a qubit. Similarly, one can send two classical bits to teleport one qubit.
Both of these require spending entanglement, so it’s not a very useful heuristic.
"one can send two classical bits to teleport one qubit"
How do you do this?
(it goes against my intuition of a qubit as state in a hilbert space)
The cost is two classical bits and one entangled Bell pair to teleport a qubit. You make the Bell pair, send one half of it to Alice (who wants to transmit the qubit) and one half to Bob (who wants to receive it). Alice then does a Bell basis measurement on the qubit she wants to transmit and the half of Bell pair she owns.
The Bell basis measurement has four outcomes. For convenience you can label them
00 01 10 11
Alice tells Bob the measurement outcome she got (this is the part that requires two bits of classical communication), and depending on which outcome she got Bob has to apply either no correction or a Pauli X, Y or Z gate.
This is known as quantum teleportation
https://en.wikipedia.org/wiki/Quantum_teleportation
and has quite a long history at this point (proposed in the early '90s and first performed in the late '90s).
Thanks, nice explanation.
"However, by taking advantage of two classical bits and an entangled qubit pair"
Yes one needs entanglement.
https://qiskit.org/textbook/ch-algorithms/teleportation.html