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

TL;DR

The paper analyzes deterministic identification over memoryless channels with finite output alphabets and arbitrary inputs, revealing that the maximal DI code size scales slightly superlinearly as $N(n,\lambda_1,\lambda_2) \sim 2^{R n \log n}$ and that the optimal rate $R$ is bounded by the lower and upper Minkowski dimensions of a transformed output set: $\frac{1}{4}\underline{d}_M(\sqrt{\widetilde{\mathcal{X}}}) \leq \dot{C}_{\text{DI}}(W) \leq \frac{1}{2}\underline{d}_M(\sqrt{\widetilde{\mathcal{X}}})$, with optimistic bounds using the upper dimension. A Hypothesis Testing Lemma shows that pairwise distinguishability of output distributions suffices to construct DI codes, and a natural spherisation-based metric on product distributions clarifies the geometry of codewords. The work also demonstrates remarkable phenomena like superactivation of DI capacity, both in classical and in classical-quantum channels, and extends the results to quantum channels under tensor-product input restrictions. Subsections provide concrete bounds for Bernoulli and Poisson channels and explore dimension-zero subtleties, continuous/additive-noise examples, and quantum extensions, culminating in a cohesive framework linking fractal geometry to finite-output DI performance and identifying several open questions for further study.

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.

Figures (5)

• Figure 1: Alice encodes a message $m$ chosen from a set $\mathcal{M}=\{1,\ldots,M\}$ into a code word of block length $n$ and sends it through $n$ uses of a noisy memoryless channel $W$. In the usual transmission scheme (above), when Bob receives $y^n$ he can decode the message, aiming to recover some $\widehat{m}\approx m$. In an identification scheme (below), he instead chooses any message $m'\in\mathcal{M}$ and checks whether it is equal to $m$ with a particular hypothesis test, obtaining a binary answer.
• Figure 2: Iterative construction of the Koch fractal. In each iteration, a triangle bend is added to each side of the current iteration, all of which are hence polygons, though the limit is not.
• Figure 3: Packing in a cube of side $s$ ($s=1$ in the direct part), and an extension containing all balls. In the converse, $s=\sqrt{2}$.
• Figure 4: Projection of two unit vectors $\sqrt{B_x}$ and $\sqrt{B_{x'}}$ onto the tangent line $\sqrt{2}-x$, and the extended vectors $\widetilde{B}_x$ and $\widetilde{B}_{x'}$.
• Figure 5: An $r$-ball packing with centers in the big sphere $S(n,\sqrt{nE})$ imposed by the power constraint, and the extended sphere $S(n,\sqrt{nE}+r)$ that contains all.

Theorems & Definitions (43)

• Definition 1.1
• Theorem 1.2: Shannon Shannon:TheoryCommunication, Wolfowitz Wolfowitz:converse
• Definition 1.3
• Theorem 1.4: Ahlswede/Dueck AD:ID_ViaChannels, Han/Verdú HanVerdu:ID
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof