Speaker
Description
A fundamental problem of inference is that of the observation of a long (ideally infinite) stationary time series of events, generated by a hidden Markov chain. What can we say about the internal structure of the hidden Markov model, aka the latent variables? If the system generating the observations is classical, we are looking to reconstruct the hidden" Markov chain from its "visible" image.
Here, we are studying the case that the hidden system is quantum mechanical, giving rise to a special class of finitely correlated states, which we call quantum hidden Markov models; 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 an associated canonical vector space with associated positive cone preserved under the hidden dynamics of the model. For the quantum case, we describe the structure of the possible GPTs via semidefinite representable (SDR) cones. It turns out that these GPTs are all finitely presented operator systems, i.e. induced subspaces of quotients of B(H) for a finite-dimensional Hilbert space H. Unlike operator systems, for which complete positivity can be very hard to decide, the SDR models come with a subset of the completely positive maps, which is itself an SDR cone [1].
We also 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 quantum hidden Markov model with finite state space [2], answering a question of Fannes, Nachtergaele and Werner [4]. Processes generated via a finite-dimensional GPT which cannot be reproduced by a classical hidden Markov chain had been known before [1,3].
References:
[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",
Commun. Math. Phys. 405(2):50 (2024).
[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).
[4] M. Fannes, B. Nachtergaele & R.F. Werner, "Finitely correlated
states on quantum spin chains", Commun. Math. Phys. 144:443-490 (1992).
