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arXiv · 2402.09117

Deterministic identification over channels with finite output: a dimensional perspective on superlinear rates

Pau Colomer·Christian Deppe·Holger Boche·Andreas Winter
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Abstract

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 as $R\,n\log n$ with 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: $\frac14 d \leq R \leq \frac12 d$. 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.

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BibTeXRIS

Pau Colomer, Christian Deppe, Holger Boche, Andreas Winter. 2024-02-14. Deterministic identification over channels with finite output: a dimensional perspective on superlinear rates. https://doi.org/10.1109/tit.2025.3531301

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