Conclusive Identification Via Noisy Classical Channel: Superactivation and Quantum Advantage

2026-03-31

AI Summary

This study addresses conclusive identification over noisy classical channels, where the receiver may output “reject” to ensure error-free input recognition. Leveraging tools from graph theory—specifically support graphs, chromatic number, and orthogonal rank—as well as channel tensor products and quantum information theory, the work demonstrates that zero-capacity channels can be activated under classical or quantum assistance. Key contributions include the first discovery of superactivation in the conclusive identification exponent; a proof that quantum assistance strictly outperforms classical assistance whenever the orthogonal rank of the support graph is less than its chromatic number; the establishment of a profound connection between this task and Kochen-Specker contextuality; and the construction of channels exhibiting arbitrarily large superactivation gaps, thereby identifying the support graph as the pivotal combinatorial structure governing these phenomena.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\mathrm{ci}_\circ(N)$ counts the maximum number of conclusively identifiable inputs. We show $\mathrm{ci}_\circ(N)$ exhibits a striking superactivation phenomenon: a channel with $\mathrm{ci}_\circ(N) = 0$ achieves $\mathrm{ci}_\circ(N \otimes \mathrm{id}^c_β) = |X|$ 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 $\mathrm{ci}_\circ(N \otimes \mathrm{id}^c_β) - \mathrm{ci}_\circ(\mathrm{id}^c_β)$ 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.

Problem

• conclusive identification
• superactivation
• quantum advantage
• noisy classical channel
• Kochen-Specker contextuality

Innovation

• conclusive identification
• superactivation
• quantum advantage
• support graph
• contextuality