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Learning a hidden property of a quantum system typically requires a series ofinteractions. In this work, we formalise such multi-round learning processesusing a generalisation of classical-quantum states, called classical-quantumcombs. Here, "classical" refers to a random variable encoding the hiddenproperty to be learnt, and "quantum" refers to the quantum comb describing thebehaviour of the system. The optimal strategy for learning the hidden propertycan be quantified by applying the comb min-entropy (Chiribella and Ebler, NJP,2016) to classical-quantum combs. To demonstrate the power of this approach, wefocus attention on an array of problems derived from measurement-based quantumcomputation (MBQC) and related applications. Specifically, we describe a knownblind quantum computation (BQC) protocol using the combs formalism and therebyleverage the min-entropy to provide a proof of single-shot security formultiple rounds of the protocol, extending the existing result in theliterature. Furthermore, we consider a range of operationally motivatedexamples related to the verification of a partially unknown MBQC device. Theseexamples involve learning the features of the device necessary for its correctuse, including learning its internal reference frame for measurementcalibration. We also introduce a novel connection between MBQC and quantumcausal models that arises in this context.

The Min-Entropy of Classical-Quantum Combs for Measurement-Based Applications