Conclusive Identification Via Noisy Classical Channel: Superactivation and Quantum Advantage

Abstract: 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 $N : X \to X$, the single-shot conclusive identification index $\mathrm{ci}<em>\circ(N)$ counts the maximum number of conclusively identifiable inputs. We show $\mathrm{ci}</em>\circ(N)$ exhibits a striking superactivation phenomenon: a channel with $\mathrm{ci}<em>\circ(N) = 0$ achieves $$\mathrm{ci}</em>\circ(N \otimes \mathrm{id}^c_β) = |X|$$ when assisted by a perfect classical channel of dimension $β&lt; |X|$. The minimum classical assistance required equals the chromatic number $χ(\mathtt{S}<em>N)$ of the channel's support graph $\mathtt{S}_N$. We provide channel families where the superactivation gap $$\mathrm{ci}</em>\circ(N \otimes \mathrm{id}^c_β) - \mathrm{ci}<em>\circ(\mathrm{id}^c</em>β)$$ 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 $ξ(\mathtt{S}_N) &lt; χ(\mathtt{S}_N)$. 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.

• The paper introduces conclusive identification, a single-shot task requiring zero false identifications while allowing inconclusive outcomes, and shows that the support graph—not the confusability graph—determines performance.
• The paper proves that an SNFC channel achieves full identification with classical assistance exactly when its assistance dimension reaches the support graph’s chromatic number, producing unbounded superactivation gaps for wheel, friendship, star, and Turán graph families.
• The paper demonstrates quantum advantage when orthogonal rank is below chromatic number, including contextuality-based examples and Newman-graph constructions yielding an exponential classical-to-quantum assistance separation.

The conclusive identification task

The paper introduces a new single-shot communication task for classical channels, termed conclusive identification. Given an XY-equivalent channel $N : X \to X$ (input and output alphabets coincide), the receiver must identify the transmitted input without error whenever possible, but is permitted to respond inconclusively when the output is ambiguous. The relevant figure of merit is the conclusive identification index $\mathrm{ci}_\circ(N)$, the maximum number of inputs that can be conclusively identified with zero probability of false identification. This places the task between Shannon's zero-error framework, where every message must be decoded, and trivial abstention; it is formally analogous to unambiguous state discrimination, with the second-kind error constrained to be exactly zero while the first-kind error may be as large as one. It differs from the Ahlswede–Dueck identification framework in operating at zero error of the second kind and in the single-shot regime.

A central structural claim is that the appropriate combinatorial object for this task is not the confusability graph $\mathtt{G}_N$, which governs zero-error capacity, but the support graph $\mathtt{S}_N$, whose edges encode the zero-nonzero pattern of the channel matrix. For symmetric not-fully-corrupted (SNFC) channels — those with $P(x|x) > 0$ for all $x$ and symmetric zero-pattern — the confusability graph is determined by the nonzero pattern of $\mathbb{S}_N^2$: two inputs are confusable exactly when they share a common neighbor in $\mathtt{S}_N$. Consequently, many distinct support graphs collapse onto the same confusability graph: the paper exhibits six SNFC channels on five symbols whose confusability graph is $\mathtt{K}_5$, yet whose support graphs differ. All such channels have $c_\circ(N) = 0$, zero entanglement-assisted zero-error capacity, and are inert under noiseless assistance in the zero-error setting — properties (P1)–(P4) that would normally classify them as useless.

Classical assistance and superactivation

The first main result is an exact characterization of the classical assistance required for full conclusive identification. For an SNFC channel $\mathrm{ci}_\circ(N)$0 with $\mathrm{ci}_\circ(N)$1, $\mathrm{ci}_\circ(N)$2, and $\mathrm{ci}_\circ(N)$3, a noiseless classical channel $\mathrm{ci}_\circ(N)$4 achieves $\mathrm{ci}_\circ(N)$5 if and only if $\mathrm{ci}_\circ(N)$6, the chromatic number of the support graph. Sufficiency follows because a proper coloring partitions the input alphabet into classes within which each input has a private output; necessity follows because any successful encoding induces a proper coloring. This yields superactivation: channels with $\mathrm{ci}_\circ(N)$7 satisfy $\mathrm{ci}_\circ(N)$8 whenever $\mathrm{ci}_\circ(N)$9.

The superactivation gap $\mathtt{G}_N$0 is shown to grow without bound. For support graph $\mathtt{G}_N$1, the gap is $\mathtt{G}_N$2 (even $\mathtt{G}_N$3) or $\mathtt{G}_N$4 (odd $\mathtt{G}_N$5); however, for $\mathtt{G}_N$6 these channels have nonzero zero-error capacity $\mathtt{G}_N$7, so they are not fully useless in Shannon's sense. To address this, the paper proves that wheel-graph channels ($\mathtt{G}_N$8, diameter 2, hence confusability graph $\mathtt{G}_N$9) have both $\mathtt{S}_N$0 and $\mathtt{S}_N$1, yet achieve $\mathtt{S}_N$2, unbounded in $\mathtt{S}_N$3. Friendship, star, and Turán graph families exhibit the same behavior. A notable fine-grained observation: two support graphs obtained by adding three chords to $\mathtt{S}_N$4 require different amounts of assistance ($\mathtt{S}_N$5 versus $\mathtt{S}_N$6), traced to degree-sequence asymmetry — evidence that the support graph captures operational distinctions invisible to the confusability graph.

Quantum assistance via orthogonal rank

The quantum counterpart replaces chromatic number by orthogonal rank. If $\mathtt{S}_N$7 is an orthogonal representation of $\mathtt{S}_N$8 in $\mathtt{S}_N$9, Alice sends $P(x|x) > 0$0 through a noiseless $P(x|x) > 0$1-dimensional quantum channel alongside the noisy transmission; Bob measures $P(x|x) > 0$2 conditioned on the classical output $P(x|x) > 0$3. Orthogonality on edges guarantees that a corrupted input never triggers a false acceptance, so $P(x|x) > 0$4 suffices, where $P(x|x) > 0$5 is the orthogonal rank. Whenever $P(x|x) > 0$6, quantum assistance strictly outperforms optimal classical assistance.

This separation is instantiated through three constructions spanning the spectrum of Kochen–Specker contextuality:

Construction	Contextuality type	$P(x|x) > 0$7	$P(x|x) > 0$8	$P(x|x) > 0$9
$x$0 (Cabello et al.)	Combinatorial state-independent	18	5	4
$x$1 (Yu–Oh)	Algebraic state-independent	13	4	3
$x$2	State-dependent	14	5	4

All three support graphs have diameter 2, so the corresponding channels have confusability graph $x$3 and both $x$4. In each case a quantum channel of dimension $x$5 accomplishes what requires a classical channel of dimension $x$6. An instructive subtlety is noted for the Cabello construction: replacing the full orthogonality graph by the sparser clique-completion of its context hypergraph preserves the advantage, indicating that the context structure rather than the complete edge set drives the phenomenon. More generally, since $x$7 characterizes state-independent contextuality for orthogonality graphs of projector sets (2604.00089), any such graph with diameter two and no isolated vertices yields a channel exhibiting both superactivation and quantum advantage.

Scaling of the advantage

Two amplification mechanisms are established. First, under the co-normal product, the ratio $x$8 satisfies $x$9, using multiplicativity of fractional chromatic number and submultiplicativity of orthogonal rank; the diameter-two property is preserved, so the product channels retain $\mathbb{S}_N^2$0. Second, and more strikingly, Newman-graph channels achieve exponential advantage directly: for $\mathbb{S}_N^2$1, the Newman graph has $\mathbb{S}_N^2$2 vertices, diameter 2, orthogonal rank $\mathbb{S}_N^2$3, and independence number bounded via Frankl–Rödl by $\mathbb{S}_N^2$4, giving

$\mathbb{S}_N^2$5

an exponential separation between required classical and quantum assistance dimensions. The paper emphasizes that this contradicts no known principle: Holevo's bound and its single-shot refinement concern simulation of channels within Shannon's framework, whereas here the quantum channel serves as assistance to a task outside that framework, yielding an advantage even without shared entanglement.

Limitations and open questions

Several caveats bear on the strength of the results. The task is defined only for SNFC channels, a restrictive symmetry condition on the channel matrix; extension to general channels or to quantum channels is open. All three explicit contextuality-based examples achieve only a unit gap $\mathbb{S}_N^2$6; whether larger finite gaps exist, and what the maximum gap is for fixed $\mathbb{S}_N^2$7, remains unresolved. The co-normal product scaling is proved only as a lower bound, and exact multiplicativity of $\mathbb{S}_N^2$8 is unknown. The asymptotic index $\mathbb{S}_N^2$9 is defined but not analyzed. The roles of shared entanglement and of bounded-error variants are left unexplored, as is a quantum analogue of the support graph for quantum channels. Finally, the exponential advantage rests on the Frankl–Rödl independence bound for Hadamard-type graphs, an analytic rather than exact estimate.

Conclusion

The paper establishes conclusive identification as a communication task in which the support graph, not the confusability graph, is the operative combinatorial invariant: minimum classical assistance equals $\mathtt{S}_N$0, minimum quantum assistance is at most $\mathtt{S}_N$1, and the separation between these quantities — witnessed by combinatorial, algebraic, and state-independent-to-state-dependent forms of Kochen–Specker contextuality — yields strict, scalable, and in some cases exponentially large quantum advantage. Channels dismissed as useless by Shannon's zero-error theory thereby acquire rich operational structure, providing a new interface between zero-error information theory and quantum foundations.