Seminario de Categorías

Expositor: Arthur Parzygnat (IHÉS).

Resumen: The foundations of probability, statistics, and information theory are slowly undergoing a potentially dramatic change of perspective through the language of category theory via string diagrams [3]. This point of view has been abstracted to finite-dimensional C*-algebras via a stochastic variant of the Gelfand—Naimark theorem and quantum Markov categories. Through this abstraction, one can immediately analyze concepts such as Bayesian inversion in non-classical settings, which in fact has recently been done in finite dimensions [2]. What can be said for more general (possibly infinite dimensional) C*-algebras? In this talk, I will review some background on Markov categories, provide some motivation for their study, introduce our main quantum example, and then I'll provide a small refresher on C*-tensor products. Then, I'll explain why all C*-algebras do not form a quantum Markov category, and I will provide some suggestions for an alternative framework [1].

Main reference:

[1] arxiv.org/abs/2001.08375 (especially Remark 3.12 and Question 3.25).

Additional references of potential interest:

[2] arxiv.org/abs/2005.03886 (on Bayesian inversion in quantum mechanics)

[3] arxiv.org/abs/1908.07021 (on classical Markov categories) 

https://www.youtube.com/watch?v=iZ8i3PbmcOM&feature=youtu.be