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SS 2024
24.04.2024 um 10:00 Uhr in 32/110
Tim Seynnaeve (KU Leuven)
The translation-invariant Bell polytope
Bell's theorem, which states that the predictions of quantum theory cannot be accounted for by any classical theory, is a foundational result in quantum physics. In modern language, it can be formulated as a strict inclusion between two geometric objects: the Bell polytope en the convex body of quantum behaviours. Describing these objects leads to a deeper understanding of the nonlocality of quantum theory, and has been a central research theme is quantum information theory for several decades. After giving an introduction to the topic, I will focus on the so-called translation-invariant Bell polytope. Physically, this object describes Bell inequalities of a translation-invariant system; mathematically it is obtained as a certain projection of the ordinary Bell polytope. Studying the facet inequalities of this polytopes naturally leads into the realm of tensor networks, combinatorics, and tropical algebra.
This talk is based on joint work in progress with Jordi Tura, Mengyao Hu, Eloic Vallée, and Patrick Emonts.
08.05.2024 um 10:00 Uhr in 32/110
Ben Hollering (TU München)
Hyperplane Representations of Interventional Characteristic Imset Polytopes
Characteristic imsets are 0/1-vectors representing directed acyclic graphs whose edges represent direct cause-effect relations between jointly distributed random variables. A characteristic imset (CIM) polytope is the convex hull of a collection of characteristic imsets. These polytopes arise as feasible regions of an integer linear programming approach to the problem of causal disovery, which aims to infer a cause-effect structure from data. Linear optimization methods typically require a hyperplane representation of the feasible region, which has proven difficult to compute for CIM polytopes despite continued efforts. We solve this problem for CIM polytopes that are the convex hull of the imsets associated to DAGs whose underlying graph of adjacencies is a tree. Our methods use the theory of toric fiber products as well as the novel notion of interventional CIM polytopes. We obtain our results by proving a more general result for families of interventional CIM polytopes. As a demonstration of the applications of these novel polytopes, we apply our results to a real data example where we solve a linear optimization problem to learn a causal system from a combination of observational and interventional data.
