Generalized probabilistic theories and tensor products of normed spaces

Abstract: Generalized Probabilistic Theories (GPTs) form an abstract framework to describe theories of nature that have probabilistic features. A GPT must specify the set of states purporting to represent the physical reality, the allowable measurements, the rules for outcome statistics of the latter, and the composition rules describing what happens when we merge subsystems and create a larger system. Examples include classical probability and quantum theory. The composition rules alluded to above usually involve tensor products and, in some formulations, normed spaces. Among tensor products of normed spaces that have operational meaning in the GPT context, the projective and the injective product are the extreme ones, which leads to the natural question "How much do they differ?" considered already by Grothendieck and Pisier (in the 1950s and 1980s). Surprisingly, no systematic quantitative analysis of the finite-dimensional case seems to have ever been made. We show that the projective/injective discrepancy is always lower-bounded by the power of the (smaller) dimension, with the exponent depending on the generality of the setup. Some of the results are essentially optimal, but others can be likely improved. The methods involve a wide range of techniques from geometry of Banach spaces and random matrices. Joint work with G. Aubrun, L. Lami, C. Palazuelos, A. Winter.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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