CQT Online Talks - Series: Computer Science Seminars
Speaker: Andreas Winter, Universitat Autònoma de Barcelona in Spain
Abstract: 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 generalized probabilistic theory (GPT). The latter case is entirely described in terms of the rank of the so-called Hankel matrix, 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 it-self an SDR cone. My aim is to discuss what we know about the geometry of the GPT cones and this ugly/beautiful mapping cone, and how they relate to the operator system structure on the hand hand, and to general GPTs on the other. [Based on joint work with Alex Monras, arXiv:1412.3634, and others]