Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

可以说，通过有限内存生成的最大类随机过程由通过适当的广义概率理论（GPT）中的连续测量产生的观察序列组成。它们是由在一组可能的线性映射下演化的有限维记忆构建的，并且结果的概率由记忆状态的线性函数确定。这种模型的例子是经典的隐马尔可夫过程，其中记忆状态是概率分布，并且在每一步都根据非负矩阵演化，以及隐量子马尔可夫过程，其中记忆是有限维量子系统，每一步都按照完全正向的地图演化。在这里，我们证明了承认有限维解释的一组过程不需要用经典概率或量子力学来解释。也就是说，我们展示了具有有限维解释的过程族，这些解释显然是由明确给定的 GPT 的动力学定义的，但不允许有限维中的量子解释，因此甚至不是经典解释。此外，我们提出了一系列关于量子位和量子体的量子过程，它们不承认经典的有限维实现，其中包括 Fox、Rubin、Dharmadikari 和 Nadkarni 之前介绍的作为无限维马尔可夫链函数的例子，以及下界实现量子位过程的噪声版本的经典模型的内存大小。