Following initial work by JaJa and Ahlswede/Cai, and inspired by a recent renewed surge in interest in deterministic identification 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 (the closure of) the subset of its output distributions in the probability simplex. Our main findings are that the maximum number of messages thus identifiable scales super-exponentially as $2^{R\,n\log n}$ with the block length $n$, and that the optimal rate $R$ is upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension $d$ of the output set: $\frac14 d \leq R \leq d$. Leading up to the general case, we treat the important special case of the so-called Bernoulli channel with input alphabet $[0;1]$ and binary output, which has $d=1$, to gain intuition. Along the way, we show a certain Hypothesis Testing Lemma (generalising an earlier insight of Ahlswede regarding the intersection of typical sets) that implies that for the construction of a deterministic identification code, it is sufficient to ensure pairwise reliable distinguishability of the output distributions. These results are then shown to generalise directly to classical-quantum channels with finite-dimensional output quantum system (but arbitrary input alphabet), and 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和Ahlswede/Cai的早期工作，并受近期通过噪声信道进行确定性识别研究再度兴起的启发，我们考虑有限输出无记忆信道（但输入字母表任意）的一般性问题。此类信道本质上由其输出分布在概率单纯形中的子集（闭包）所决定。我们的主要发现是：可识别的最大消息数随分组长度$n$呈超指数增长$2^{R\,n\log n}$，且最优速率$R$受输出集覆盖维数（又称闵可夫斯基维数、柯尔莫哥洛夫维数或熵维数）$d$的上下界约束：$\frac14 d \leq R \leq d$。为过渡至一般情形，我们首先处理重要的特例——伯努利信道（输入字母表$[0;1]$，二进制输出，具有$d=1$）以获取直观认识。在此过程中，我们提出了一个假设检验引理（推广了Ahlswede关于典型集交集的早期洞见），该引理表明：构建确定性识别码时，只需确保输出分布两两可靠可区分。继而证明这些结果可直接推广至有限维输出量子系统（但输入字母表任意）的经典-量子信道，特别适用于识别码仅能使用张量积输入的有限维量子系统上的量子信道。