Conclusive Identification Via Noisy Classical Channel: Superactivation and Quantum Advantage

We introduce conclusive identification task for classical channels: a receiver identifies transmitted inputs without error when possible, and responds inconclusively when outputs are ambiguous. For a symmetric not-fully-corrupted channel N : X → X, the single-shot conclusive identification index ci_circ(N) counts the maximum number of conclusively identifiable inputs. We show ci_circ(N) exhibits a striking superactivation phenomenon: a channel with ci_circ(N) = 0 achieves ci_circ\(N otimes id^c_β\) = |X| when assisted by a perfect classical channel of dimension β< |X|. The minimum classical assistance required equals the chromatic number χ\(mathtt{S}_N\) of the channel's support graph mathtt{S}_N. We provide channel families where the superactivation gap ci_circ\(N otimes id^c_β\) - ci_circ\(id^c_β\) can be made arbitrarily large. A noiseless quantum channel of dimension equal to the orthogonal rank ξ\(mathtt{S}_N\) suffices, yielding a strict quantum advantage whenever ξ\(mathtt{S}_N\) < χ\(mathtt{S}_N\). This advantage is demonstrated through three explicit constructions motivated by combinatorial and algebraic state-independent, and state-dependent proofs of Kochen-Specker contextuality. Via the co-normal product of graphs, we analyze the scaling of the quantum advantage ratio χ_f\(mathtt{S}_N\)/ξ\(mathtt{S}_N\), and present a channel for which quantum assistance is exponentially more efficient than classical. Our results establish mathtt{S}_N, rather than the confusability graph mathtt{G}_N, as the natural combinatorial object for conclusive identification, revealing that channels deemed useless under Shannon's zero-error framework can exhibit rich superactivation and quantum advantage, with deep connections to quantum contextuality.