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

TL;DR

This paper introduces a framework using classical-quantum combs and min-entropy to analyze multi-round quantum learning processes, with applications to measurement-based quantum computation and security verification.

Contribution

It formalizes multi-round quantum learning using classical-quantum combs and applies min-entropy to analyze security and device verification in measurement-based quantum computation.

Findings

Proves single-shot security for blind quantum computation protocols.

Connects MBQC with quantum causal models.

Provides methods for learning device features and internal reference frames.

Abstract

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.