Bounds and guarantees for learning and entanglement

• URL: http://arxiv.org/abs/2404.07277v1
• Date: Wed, 10 Apr 2024 18:09:22 GMT
• Title: Bounds and guarantees for learning and entanglement
• Authors: Evan Peters,
• Abstract summary: Information theory provides tools to predict the performance of a learning algorithm on a given dataset. This work first extends this relationship by demonstrating that a small conditional entropy is also sufficient for successful learning. This connection between information theory and learning suggests that we might similarly apply quantum information theory to characterize learning tasks involving quantum systems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Information theory provides tools to predict the performance of a learning algorithm on a given dataset. For instance, the accuracy of learning an unknown parameter can be upper bounded by reducing the learning task to hypothesis testing for a discrete random variable, with Fano's inequality then stating that a small conditional entropy between a learner's observations and the unknown parameter is necessary for successful estimation. This work first extends this relationship by demonstrating that a small conditional entropy is also sufficient for successful learning, thereby establishing an information-theoretic lower bound on the accuracy of a learner. This connection between information theory and learning suggests that we might similarly apply quantum information theory to characterize learning tasks involving quantum systems. Observing that the fidelity of a finite-dimensional quantum system with a maximally entangled state (the singlet fraction) generalizes the success probability for estimating a discrete random variable, we introduce an entanglement manipulation task for infinite-dimensional quantum systems that similarly generalizes classical learning. We derive information-theoretic bounds for succeeding at this task in terms of the maximal singlet fraction of an appropriate finite-dimensional discretization. As classical learning is recovered as a special case of this task, our analysis suggests a deeper relationship at the interface of learning, entanglement, and information.

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