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.

翻译：继JaJa、Ahlswede和Cai的奠基性工作之后，并受到近期对噪声信道确定性识别研究兴趣复兴的启发，我们在一般性框架下研究该问题：针对具有有限输出字母表但输入字母表任意的无记忆信道。此类信道本质上由其输出分布（作为概率单纯形中的一个子集）所刻画。我们的主要发现是：可识别消息的最大长度随码块长度$n$呈超线性增长，其尺度为$R\,n\log n$；且最优速率$R$可由输出集经特定代数变换后的覆盖维数（亦称Minkowski维数、Kolmogorov维数或熵维数）$d$界定：$\frac14 d \leq R \leq \frac12 d$。值得注意的是，Minkowski维数的下界与上界在此结果中均发挥作用。在推导过程中，我们提出了一个"假设检验引理"，证明只需确保输出分布之间具备两两可靠可区分性，即可构建确定性识别码。尽管尚未得到精确的容量公式，我们可以得出结论：确定性识别容量具有超激活特性——存在这样的信道，其各自容量为零，但它们的直积信道却具有正容量。我们还将这些结果推广至具有有限维输出量子系统的经典-量子信道，特别是针对有限维量子系统上的量子信道，并约束识别码仅允许使用张量积输入。