Associative memories are devices storing information that can be fully retrieved given partial disclosure of it. We examine a toy model of associative memory and the ultimate limitations it is subjected to within the framework of general probabilistic theories (GPTs), which represent the most general class of physical theories satisfying some basic operational axioms. We ask ourselves how large the dimension of a GPT should be so that it can accommodate $2^m$ states with the property that any $N$ of them are perfectly distinguishable. Call $d(N,m)$ the minimal such dimension. Invoking an old result by Danzer and Gr\"unbaum, we prove that $d(2,m)=m+1$, to be compared with $O(2^m)$ when the GPT is required to be either classical or quantum. This yields an example of a task where GPTs outperform both classical and quantum theory exponentially. More generally, we resolve the case of fixed $N$ and asymptotically large $m$, proving that $d(N,m) \leq m^{1+o_N(1)}$ (as $m\to\infty$) for every $N\geq 2$, which yields again an exponential improvement over classical and quantum theories. Finally, we develop a numerical approach to the general problem of finding the largest $N$-wise mutually distinguishable set for a given GPT, which can be seen as an instance of the maximum clique problem on $N$-regular hypergraphs.

翻译：联想记忆是一种存储信息的设备，能在仅获得部分信息的情况下完整检索内容。我们研究了一个联想记忆的玩具模型，以及它在通用概率理论（GPTs）框架下所受的终极限制——该理论代表了满足若干基本操作公理的最一般物理理论体系。我们提出了一个核心问题：GPT的维度需要多大，才能容纳$2^m$个状态，且其中任意$N$个状态可以完美区分？记$d(N,m)$为所需的最小维度。通过引用Danzer和Grünbaum的经典结论，我们证明了$d(2,m)=m+1$，而经典或量子GPT对应的维度为$O(2^m)$。这表明存在某种任务，GPT能以指数级优势超越经典与量子理论。更一般地，我们解决了固定$N$且$m$渐近大时的情形，证明对于所有$N\geq 2$，有$d(N,m) \leq m^{1+o_N(1)}$（当$m\to\infty$），这同样在经典与量子理论基础上实现指数级改进。最后，我们开发了一种数值方法，用于求解给定GPT中最大N元互可区分状态集的一般性问题——该问题可看作N正则超图上的最大团问题实例。