Bounds and guarantees for learning and entanglement (2404.07277v1)

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.