Can finite correlation in a quantum spin chain always be explained by a finite quantum memory?

The answer is: no, there are states of quantum spin chains which are generated by an abstract finite memory, but not with any finite quantum memory. But to explain it, we have to start with finitely-correlated states [1]: these are (for simplicity: translation-invariant) states on an infinite quantum spin chain that are produced by a memory system, given by a finite-dimensional vector space, interacting linearly and one by one with the quantum spins, initially prepared in identical pure states (Fig. 1).Here we look at the special case of stochastic processes, generated by the memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by a linear function of the memory state. It is a fact that every bona fide finitely correlated process is related to the state space of a general probabilistic theory (GPT), which can be thought of a convex cone in the memory vector space. Examples include classical hidden Markov models (HMM), where the memory is a probability distribution, evolving at each step according to a non-negative matrix, and hidden quantum Markov models (HQMM), where the memory is a quantum state, and at each step it evolves according to a completely positive map.In [2], finitely correlated processes are exhibited which are however not realisable by HMMs. The same was shown via a novel geometric method in [3], by constructing a HQMM with a qubit memory, whose process cannot be reproduced by any HMM. Incidentally, we show in [4] that the family of processes from [2] are all realised by HQMMs. Finally, in [4] we construct a finitely correlated process that via the same geometric method is impossible to reproduce by HQMMs; this resolves an open question from [1].The key to all this is the analysis of the cone of the GPT state space, which has to be stable under the generating maps of the process. Usually there is some freedom in its choice, but in [3,4] it was forced to be unique. Furthermore, in [4] it is the exponential cone (Fig. 2), which is transcendental. However, in [3] we already observed that to support an HQMM, the cone would have to be semi-algebraic.