Andreas Winter | Hidden Markov models – classical and quantum mechanisms, and beyond

▶Title: Hidden Markov models – classical and quantum mechanisms, and beyond
▶Speaker: Andreas Winter (Universitat Autònoma de Barcelona)
▶Abstract: Among the stationary time series of events (in discrete time steps and with signals from a discrete alphabet), those generated as a function of a hidden Markov chain have a particular significance. These are special cases of finitely correlated states [0] in the case that the memory system and the spins of the infinite chain are classical. Here we investigate how much more descriptive power for a process we gain by having a quantum memory systems, and even more generally, a general probabilistic theory (GPT).
The latter case is entirely described in terms of the rank of the so-called Hankel matrix of the process, and equivalently an associated canonical finite-dimensional vector space with associated positive cone preserved under the hidden dynamics of the model. For the quantum case, we can describe the structure of the possible GPTs via semidefinite representable (SDR) cones, while in the classical case the cones have to be polyhedral [1].
I will then describe the first known example of a process generated via a finite-dimensional GPT as the hidden system, which however cannot be reproduced by any hidden classical or quantum Markov model with finite state space [2], answering a question from [0]. Processes generated via a finite-dimensional GPT which cannot be reproduced by a hidden classical Markov chain had been known before [1,3], but the examples exhibited there turn out to be generated by hidden quantum Markov models.

[0] M. Fannes, B. Nachtergaele & R.F. Werner, ""Finitely correlated states on quantum spin chains"", Commun. Math. Phys. 144:443-490 (1992).
[1] A. Monràs & A. Winter, “Quantum learning of classical stochastic processes: the completely positive realization problem”, J. Math. Phys. 57:015219 (2016).
[2] M. Fanizza, J. Lumbreras & A. Winter, ""Quantum theory in finite dimension cannot explain every general process with finite memory"", arXiv:2209.11225 (2022).
[3] S.W. Dharmadhikari & M.G. Nadkarni, “Some regular and non-regular functions of finite Markov chains”, Ann. Math. Stat. 41(1):207-213
(1970).

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