Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conclusive Identification Via Noisy Classical Channel: Superactivation and Quantum Advantage

Anushko Chattopadhyay, Ambuj, Rakesh Das, Smritikana Patra, Chitrak Roychowdhury, Manik Banik, Amit Mukherjee

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.

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 , the single-shot conclusive identification index counts the maximum number of conclusively identifiable inputs. We show exhibits a striking superactivation phenomenon: a channel with achieves when assisted by a perfect classical channel of dimension . The minimum classical assistance required equals the chromatic number of the channel's support graph . We provide channel families where the superactivation gap can be made arbitrarily large. A noiseless quantum channel of dimension equal to the orthogonal rank suffices, yielding a strict quantum advantage whenever . 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 , and present a channel for which quantum assistance is exponentially more efficient than classical. Our results establish , rather than the confusability graph , 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.

Paper Structure

This paper contains 17 sections, 14 theorems, 38 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Results with classical channels
  4. Setup and Preliminaries
  5. Conclusive Identification Task
  6. Conclusive Identification with Forward Classical Assistance
  7. Superactivation of Conclusive Identification Index
  8. Arbitrarily Large Superactivation
  9. Quantum Assistance in Conclusive Identification
  10. KS Advantage in Conclusive Identification
  11. Beyond KS (system) advantage in Conclusive Identification
  12. Advantage Through State Independent Contextuality of Yu and Oh
  13. Advantage Through State Dependent Contextuality
  14. Scaling the Quantum Advantage
  15. Polynomial Scaling of Quantum Advantage
  16. ...and 2 more sections

Key Result

Proposition 1

For an SNFC channel $N : X \to X$ with support graph adjacency matrix $\mathbb{S}_N = (S_{xx'})$, the adjacency matrix $\mathbb{G}_N = (G_{xx'})$ of the confusability graph $\mathtt{G}_N$ is given by the non-zero pattern of $\mathbb{S}_N^2$, i.e. or equivalently, $G_{xx'} \neq 0 \iff \sum_{\tilde{x} \in X} S_{x\tilde{x}}\, S_{\tilde{x}x'} > 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Bipartite hypergraph representing the input domain and output range of the channel $N:\{a,b,c\}\to\{1,2\}$ defined in Eq. (\ref{['Ex-Sup']}).
  • Figure 2: Weighted digraph corresponding to the XY-equivalent channel of Eq. (\ref{['digraph']}). A directed edge starting from a vertex $x$ ends in the output-range $\Gamma_x$ of that input. A directed edge ending at a vertex $x$ originates from the input-domain $\Omega_x$ of that output. Each edge is weighted by a transition probability in $(0, 1]$, denoting the probability of transmitting that particular input to that particular output.
  • Figure 3: Support graphs $\mathtt{S}_i$ (self-loops omitted) arranged clockwise with increasing number of chords ($\mathtt{S}_4$ and $\mathtt{S}_5$ have same number of chords), all leading to the same confusability graph $\mathtt{G}_{N_i}=\mathtt{K}_5$.
  • Figure 4: (Left) Coloring of the support graph $\mathtt{S}_4$. The same coloring holds for $\mathtt{S}_3$, $\mathtt{S}_2$, and $\mathtt{S}_1$: each of these graphs has fewer edges (chords) than $\mathtt{S}_4$ and thus requires no additional colors. Although $\mathtt{S}_5$ has the same number of edges as $\mathtt{S}_4$, it requires one extra color (right). Unlike $\mathtt{S}_4$, its degree sequence is more asymmetric, making a valid coloring with the same number of colors impossible.
  • Figure 5: Wheel Graph ($\mathtt{W}_n$). For even $n$ chromatic number $\chi(\mathtt{W}_n)=3$, whereas $\chi(\mathtt{W}_n)=4$ for odd $n$. $\mathrm{diam}(\mathtt{W}_3)=1$ and for $n>3$$\mathrm{diam}(\mathtt{W}_n)=2$.
  • ...and 6 more figures

Theorems & Definitions (46)

  • Definition 1: Classical Channel
  • Definition 2: Confusability Graph
  • Definition 3: Single-shot zero-error capacity
  • Definition 4: Output-Range & Input-Domain
  • Definition 5: XY-Equivalent Channel
  • Definition 6: Fully-Corrupted Input
  • Definition 7: Symmetric Not-Fully-Corrupted Channel
  • Definition 8: Support Graph of an SNFC Channel
  • Proposition 1
  • proof
  • ...and 36 more