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Communication complexity : large output functions, partition bounds, and quantum nonlocality

Abstract : Most classical problems of communication complexity are Boolean functions. When considering functions of larger output, the way in which the result of a computation must be made available – the output model – can greatly impact the complexity of the problem. In particular, some lower bounds may not apply to all models. In this thesis, we study some lower bounds affected by the output model, problems with large outputs, revisit several classical results in the light of these output mechanisms, and relate them to the formalism of behaviors and Bell inequalities of quantum nonlocality. First, in the realm of partition bounds, we show that they necessarily have a relatively large value on large output functions, which indicates that they are lower bounds for only one of our output models. We also show how to obtain a deterministic protocol from an optimal solution of the positive partition bound, and a new connection with another lower bound technique, weak-regularity. We also leverage a recent information-theoretic re-interpretation of the partition bound to give an exponential separation between communication complexity and partition bound. The problem achieving this is a large output relation recently introduced to separate communication complexity and external information complexity. Secondly, we formally define several output models. We separate them, showing how the complexity of some problems dramatically changes between models, and for all our model re-prove standard error-reduction and randomness-removal results only previously known for the most standard, usually assumed output models. Furthermore, we show for a few natural problems that their complexity significantly varies when changing output model, and show how the rank lower bound still applies to all our models with only a slight adaptation. As last topic, we move to quantum nonlocality and show that some communication complexity lower bounds have an interpretation in the form of Bell inequalities. The Bell inequalities obtained are resistant against detection inefficiency, and a priori not against other types of disturbance. We reformulate computing a function in a specific output model as computing a behavior – a family of probability distributions indexed by possible inputs. This allows us to use the efficiency bounds as proper generalisations of the partition bound for non-standard output models.
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https://tel.archives-ouvertes.fr/tel-03342472
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Submitted on : Monday, September 13, 2021 - 1:09:25 PM
Last modification on : Wednesday, November 3, 2021 - 6:39:18 AM

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Alexandre Nolin. Communication complexity : large output functions, partition bounds, and quantum nonlocality. Computational Complexity [cs.CC]. Université de Paris, 2020. English. ⟨NNT : 2020UNIP7201⟩. ⟨tel-03342472⟩

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