Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes
This addresses the fundamental issue of determining when different models produce identical observable behavior in stochastic processes, which is crucial for researchers in quantum information and statistical modeling, though it is incremental in extending identifiability to quantum and post-quantum contexts.
The paper tackles the problem of identifiability for models of classical stochastic processes, including classical, quantum, and post-quantum models, by mapping them to a canonical generalized hidden Markov model, and it provides bounds on the minimal dimension required for quantum models to generate such processes, with some bounds being tight.
To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.