Quantum-Enhanced Zero-Error Communication and Storage under Positional Uncertainty
Permutation channels model communication and storage scenarios in which the positional identity of the physical carriers is partially or completely lost, so that the transmitted information is only accessible up to an unknown reordering. Here we show that quantum mechanics can dramatically enhance zero-error communication through such channels. For cyclic reorderings of $n$ $d$-level systems, and in the absence of positional metadata, the number of classical zero-error messages scales asymptotically as $d^n/n$, whereas quantum protocols can fully recover the identity-channel value $d^n$. Ancilla-assisted protocols further increase this number to $d^{2n}/n$, enabling dense coding under positional uncertainty. We also analyze dihedral permutation channels and derive general P\'olya-like formulas for the number of distinguishable messages in a broad class of permutation groups. Finally, for the symmetric group $S_n$, corresponding to complete scrambling of the information carriers, the number of distinguishable messages scales as $n^{d-1}$ classically, compared with $n^{d(d+1)/2-1}$ for quantum protocols and $n^{d^2-1}$ in the ancilla-assisted setting. Our results establish a fundamental quantum advantage for communication and storage under positional uncertainty.