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

[quant-ph] Learning a hidden property of a quantum system typically requires a series of interactions. In this work, we formalise such multi-round learning processes using a generalisation of classical-quantum states, called classical-quantum combs. Here, classical refers to a random variable encoding the hidden property to be learnt, and quantum refers to the quantum comb describing the behaviour of the system. The optimal strategy for learning the hidden property can be quantified by applying the comb min-entropy (Chiribella and Ebler, NJP, 2016) to classical-quantum combs. To demonstrate the power of this approach, we focus attention on an array of problems derived from measurement-based quantum computation (MBQC) and related applications. Specifically, we describe a known blind quantum computation (BQC) protocol using the combs formalism and thereby leverage the min-entropy to provide a proof of single-shot security for multiple rounds of the protocol, extending the existing result in the literature. Furthermore, we consider a range of operationally motivated examples related to the verification of a partially unknown MBQC device. These examples involve learning the features of the device necessary for its correct use, including learning its internal reference frame for measurement calibration. We also introduce a novel connection between MBQC and quantum causal models that arises in this context.

Automated summary

In this paper, we formalize the process of learning hidden properties of quantum systems using classical-quantum combs, a generalization of classical-quantum states. We quantify the optimal strategy for learning the hidden property using the comb min-entropy and demonstrate its power in the context of measurement-based quantum computation (MBQC) and verification of unknown MBQC devices. We also introduce a novel connection between MBQC and quantum causal models.