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arXiv:2604.00089

Conclusive Identification Via Noisy Classical Channel: Superactivation and Quantum Advantage

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

quant-phcs.ITSubmitted Mar 31, 2026• Announced Apr 2, 2026• 0 citations
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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.

Comments
30.25 pages (single column) + 3.5 pages (single column), 11 figures; Comments are welcome

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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→XN : X \to XN:X→X, the single-shot conclusive identification index ci∘(N)\mathrm{ci}_\circ(N)ci∘​(N) counts the maximum number of conclusively identifiable inputs. We show ci∘(N)\mathrm{ci}_\circ(N)ci∘​(N) exhibits a striking superactivation phenomenon: a channel with ci∘(N)=0\mathrm{ci}_\circ(N) = 0ci∘​(N)=0 achieves ci∘(N⊗idβc)=∣X∣\mathrm{ci}_\circ(N \otimes \mathrm{id}^c_β) = |X|ci∘​(N⊗idβc​)=∣X∣ when assisted by a perfect classical channel of dimension β<∣X∣β< |X|β<∣X∣. The minimum classical assistance required equals the chromatic number χ(SN)χ(\mathtt{S}_N)χ(SN​) of the channel's support graph SN\mathtt{S}_NSN​. We provide channel families where the superactivation gap ci∘(N⊗idβc)−ci∘(idβc)\mathrm{ci}_\circ(N \otimes \mathrm{id}^c_β) - \mathrm{ci}_\circ(\mathrm{id}^c_β)ci∘​(N⊗idβc​)−ci∘​(idβc​) can be made arbitrarily large. A noiseless quantum channel of dimension equal to the orthogonal rank ξ(SN)ξ(\mathtt{S}_N)ξ(SN​) suffices, yielding a strict quantum advantage whenever ξ(SN)<χ(SN)ξ(\mathtt{S}_N) < χ(\mathtt{S}_N)ξ(SN​)<χ(SN​). 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 χf(SN)/ξ(SN)χ_f(\mathtt{S}_N)/ξ(\mathtt{S}_N)χf​(SN​)/ξ(SN​), and present a channel for which quantum assistance is exponentially more efficient than classical. Our results establish SN\mathtt{S}_NSN​, rather than the confusability graph GN\mathtt{G}_NGN​, 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.
We demonstrate superadditivity of one-shot zero-error classical capacity in an asymmetric communication setting where a noisy classical channel is used in parallel with a perfect quantum channel. Each channel individually supports only a fixed number of perfectly distinguishable messages. Their joint use enables transmission of strictly more messages than permitted by the product of the individual capacities. We present explicit constructions achieving this enhancement and establish that replacing the perfect quantum channel with a perfect classical channel eliminates the effect. Finally, we identify a structural criterion on the noisy channel governing this effect and show that the quantum advantage is rooted in Kochen-Specker contextuality.
Classical identification via a noisy channel is a communication task in which the receiver is not required to reconstruct the full transmitted message, but only to decide whether it coincides with a message of interest. This relaxation allows the number of identifiable messages to grow doubly exponentially with the blocklength. For quantum channels, the resulting (doubly exponential) identification capacity CIDC_{\mathrm{ID}}CID​ can strictly exceed the ordinary (exponential) transmission capacity CCC. In this paper, we prove that the entanglement-assisted transmission capacity CEC_ECE​ is a strong converse bound for this task: CID≤CEC_{\mathrm{ID}}\leq C_ECID​≤CE​. For sufficiently low-noise channels, this bound can also be achieved via the Hayden-Winter (quantum) identification + fingerprinting codes. This yields an exact characterization CID=CEC_{\mathrm{ID}}=C_ECID​=CE​ of identification capacity for such channels. However, for general channels, we prove that this upper bound can be strict. We exhibit an explicit family of transpose-depolarizing channels for which CID<CEC_{\mathrm{ID}}<C_ECID​<CE​. As a consequence, we also obtain the first example of strict superadditivity of the identification capacity CIDC_{\mathrm{ID}}CID​.
We study various super-activation effects in the following zero-error communication scenario: One sender wants to send classical or quantum information through a noisy quantum channel to one receiver with zero probability of error. First we show that there are quantum channels of which a single use is not able to transmit classical information perfectly yet two uses can. This is achieved by employing entangled input states between different uses of the given channel and thus cannot happen for classical channels. Second we exhibit a class of quantum channel with vanishing zero-error classical capacity such that when a noiseless qubit channel or one ebit shared entanglement are available, it can be used to transmit log⁡2d\log_2 dlog2​d noiseless qubits, where 2d is the dimension of input state space. Third we further construct quantum channels with vanishing zero-error classical capacity when assisted with classical feedback can be used to transmit both classical and quantum information perfectly. These striking findings not only indicate both the zero-error quantum and classical capacities of quantum channels satisfy a strong super-additivity beyond any classical channels, but also highlight the activation power of auxiliary physical resources in zero-error communication.
Following initial work by JaJa, Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by its output distributions as a subset in the probability simplex. Our main findings are that the maximum length of messages thus identifiable scales superlinearly asR nlog⁡nR\,n\log nRnlognwith the block length n, and that the optimal rate R is bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension d of a certain algebraic transformation of the output set:14d≤R≤12d\frac {1}{4} d \leq R \leq \frac {1}{2} d41​d≤R≤21​d. Remarkably, both the lower and upper Minkowski dimensions play a role in this result. Along the way, we present a Hypothesis Testing Lemma showing that it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits superactivation: there exist channels whose capacities individually are zero, but whose product has positive capacity. We also generalise these results to classical-quantum channels with finite-dimensional output quantum system, in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly, so that they can be decoded with zero probability of error. The study of zero-error capacities dates right back to Shannon and the early days of information theory. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel. We further strengthen this result to show that there exist pairs of channels, neither of which have any zero-error classical capacity (as before), yet for which access to one copy of the joint channel even allows far more delicate quantum information to be transmitted perfectly. This subsumes the first result, and also implies that the quantum zero-error capacity can be superactivated. But it is strictly stronger than either of these. Indeed, this is the strongest conceivable form of superactivation, and nothing similar is possible for standard Shannon capacities of quantum channels or for zero-error capacities of classical channels.
Recently the one-shot quantum no-signalling assisted zero-error classical capacity of a quantum channel has been formulated as a semidefinite programming (SDP) depending only on the Choi-Kraus operator space of the channel. In this paper, we study the \textit{activated quantum no-signalling assisted zero-error classical capacity} by first allowing the assistance from some noiseless forward communication channel and later paying back the cost of the helper. We show that the one-shot activated capacity can also be formulated as a SDP and derive a number of striking properties of this number. In particular, this number is additive under direct sum, and is always greater than or equal to the super-dense coding bound. An a remarkable consequence, we find that one bit noiseless classical communication is able to fully activate any classical-quantum channel to achieve its asymptotic capacity, or the semidefinite fractional packing number. We also discuss the condition under which a noisy channel can activate an activatable channel. Interestingly, a channel is able to activate itself if its one-shot capacity is greater than or equal to the Golden Ratio - (1+5)/2(1+\sqrt{5})/2(1+5​)/2. We also show that the asymptotic activated capacity is still equal to the usual no-signalling assisted capacity. Finally, we show that general the asymptotic no-signalling assisted zero-error capacity does not equal to the semidefinite (fractional) packing number by an explicit construction.