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Learning tasks play an increasingly prominent role in quantum information andcomputation. They range from fundamental problems such as state discriminationand metrology over the framework of quantum probably approximately correct(PAC) learning, to the recently proposed shadow variants of state tomography.However, the many directions of quantum learning theory have so far evolvedseparately. We propose a general mathematical formalism for describing quantumlearning by training on classical-quantum data and then testing how well thelearned hypothesis generalizes to new data. In this framework, we prove boundson the expected generalization error of a quantum learner in terms of classicaland quantum information-theoretic quantities measuring how strongly thelearner's hypothesis depends on the specific data seen during training.  To achieve this, we use tools from quantum optimal transport and quantumconcentration inequalities to establish non-commutative versions of decouplinglemmas that underlie recent information-theoretic generalization bounds forclassical machine learning.  Our framework encompasses and gives intuitively accessible generalizationbounds for a variety of quantum learning scenarios such as quantum statediscrimination, PAC learning quantum states, quantum parameter estimation, andquantumly PAC learning classical functions. Thereby, our work lays a foundationfor a unifying quantum information-theoretic perspective on quantum learning.

Information-theoretic generalization bounds for learning from quantum data