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 state is a finite dimensional quantum state, 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 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.

翻译：可以说，通过有限记忆生成的最大一类随机过程，是在合适的广义概率论框架下，由连续测量产生的观测序列构成的。这些过程构造自一个有限维记忆体，它在一组可能的线性映射下演化，而结果的概率由记忆状态的线性函数决定。此类模型的例子包括经典隐马尔可夫过程（其中记忆状态为概率分布，每步按非负矩阵演化）和隐量子马尔可夫过程（其中记忆状态为有限维量子态，每步按完全正映射演化）。本文表明，允许有限维解释的过程集合并不需要从经典概率或量子力学的角度来解释。换言之，我们展示了一系列具有有限维解释的过程族——这些解释明确由给定广义概率论的动力学定义——但它们在有限维度下并不存在量子解释（因此也不存在经典解释）。此外，我们提出了一个基于量子比特和量子三维系统的量子过程族，它无法实现经典有限维表示，其中包含Fox、Rubin、Dharmadikari和Nadkarni早期引入的作为无限维马尔可夫链函数的例子，并且我们给出了实现含噪版本量子比特过程的经典模型记忆大小的下界。